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Vincent Selhorst-Jones

Vincent Selhorst-Jones

Introduction to Logarithms

Slide Duration:

Table of Contents

I. Introduction
Introduction to Precalculus

10m 3s

Intro
0:00
Title of the Course
0:06
Different Names for the Course
0:07
Precalculus
0:12
Math Analysis
0:14
Trigonometry
0:16
Algebra III
0:20
Geometry II
0:24
College Algebra
0:30
Same Concepts
0:36
How do the Lessons Work?
0:54
Introducing Concepts
0:56
Apply Concepts
1:04
Go through Examples
1:25
Who is this Course For?
1:38
Those Who Need eExtra Help with Class Work
1:52
Those Working on Material but not in Formal Class at School
1:54
Those Who Want a Refresher
2:00
Try to Watch the Whole Lesson
2:20
Understanding is So Important
3:56
What to Watch First
5:26
Lesson #2: Sets, Elements, and Numbers
5:30
Lesson #7: Idea of a Function
5:33
Lesson #6: Word Problems
6:04
What to Watch First, cont.
6:46
Lesson #2: Sets, Elements and Numbers
6:56
Lesson #3: Variables, Equations, and Algebra
6:58
Lesson #4: Coordinate Systems
7:00
Lesson #5: Midpoint, Distance, the Pythagorean Theorem and Slope
7:02
Lesson #6: Word Problems
7:10
Lesson #7: Idea of a Function
7:12
Lesson #8: Graphs
7:14
Graphing Calculator Appendix
7:40
What to Watch Last
8:46
Let's get Started!
9:48
Sets, Elements, & Numbers

45m 11s

Intro
0:00
Introduction
0:05
Sets and Elements
1:19
Set
1:20
Element
1:23
Name a Set
2:20
Order The Elements Appear In Has No Effect on the Set
2:55
Describing/ Defining Sets
3:28
Directly Say All the Elements
3:36
Clearly Describing All the Members of the Set
3:55
Describing the Quality (or Qualities) Each member Of the Set Has In Common
4:32
Symbols: 'Element of' and 'Subset of'
6:01
Symbol is ∈
6:03
Subset Symbol is ⊂
6:35
Empty Set
8:07
Symbol is ∅
8:20
Since It's Empty, It is a Subset of All Sets
8:44
Union and Intersection
9:54
Union Symbol is ∪
10:08
Intersection Symbol is ∩
10:18
Sets Can Be Weird Stuff
12:26
Can Have Elements in a Set
12:50
We Can Have Infinite Sets
13:09
Example
13:22
Consider a Set Where We Take a Word and Then Repeat It An Ever Increasing Number of Times
14:08
This Set Has Infinitely Many Distinct Elements
14:40
Numbers as Sets
16:03
Natural Numbers ℕ
16:16
Including 0 and the Negatives ℤ
18:13
Rational Numbers ℚ
19:27
Can Express Rational Numbers with Decimal Expansions
22:05
Irrational Numbers
23:37
Real Numbers ℝ: Put the Rational and Irrational Numbers Together
25:15
Interval Notation and the Real Numbers
26:45
Include the End Numbers
27:06
Exclude the End Numbers
27:33
Example
28:28
Interval Notation: Infinity
29:09
Use -∞ or ∞ to Show an Interval Going on Forever in One Direction or the Other
29:14
Always Use Parentheses
29:50
Examples
30:27
Example 1
31:23
Example 2
35:26
Example 3
38:02
Example 4
42:21
Variables, Equations, & Algebra

35m 31s

Intro
0:00
What is a Variable?
0:05
A Variable is a Placeholder for a Number
0:11
Affects the Output of a Function or a Dependent Variable
0:24
Naming Variables
1:51
Useful to Use Symbols
2:21
What is a Constant?
4:14
A Constant is a Fixed, Unchanging Number
4:28
We Might Refer to a Symbol Representing a Number as a Constant
4:51
What is a Coefficient?
5:33
A Coefficient is a Multiplicative Factor on a Variable
5:37
Not All Coefficients are Constants
5:51
Expressions and Equations
6:42
An Expression is a String of Mathematical Symbols That Make Sense Used Together
7:05
An Equation is a Statement That Two Expression Have the Same Value
8:20
The Idea of Algebra
8:51
Equality
8:59
If Two Things Are the Same *Equal), Then We Can Do the Exact Same Operation to Both and the Results Will Be the Same
9:41
Always Do The Exact Same Thing to Both Sides
12:22
Solving Equations
13:23
When You Are Asked to Solve an Equation, You Are Being Asked to Solve for Something
13:33
Look For What Values Makes the Equation True
13:38
Isolate the Variable by Doing Algebra
14:37
Order of Operations
16:02
Why Certain Operations are Grouped
17:01
When You Don't Have to Worry About Order
17:39
Distributive Property
18:15
It Allows Multiplication to Act Over Addition in Parentheses
18:23
We Can Use the Distributive Property in Reverse to Combine Like Terms
19:05
Substitution
20:03
Use Information From One Equation in Another Equation
20:07
Put Your Substitution in Parentheses
20:44
Example 1
23:17
Example 2
25:49
Example 3
28:11
Example 4
30:02
Coordinate Systems

35m 2s

Intro
0:00
Inherent Order in ℝ
0:05
Real Numbers Come with an Inherent Order
0:11
Positive Numbers
0:21
Negative Numbers
0:58
'Less Than' and 'Greater Than'
2:04
Tip To Help You Remember the Signs
2:56
Inequality
4:06
Less Than or Equal and Greater Than or Equal
4:51
One Dimension: The Number Line
5:36
Graphically Represent ℝ on a Number Line
5:43
Note on Infinities
5:57
With the Number Line, We Can Directly See the Order We Put on ℝ
6:35
Ordered Pairs
7:22
Example
7:34
Allows Us to Talk About Two Numbers at the Same Time
9:41
Ordered Pairs of Real Numbers Cannot be Put Into an Order Like we Did with ℝ
10:41
Two Dimensions: The Plane
13:13
We Can Represent Ordered Pairs with the Plane
13:24
Intersection is known as the Origin
14:31
Plotting the Point
14:32
Plane = Coordinate Plane = Cartesian Plane = ℝ²
17:46
The Plane and Quadrants
18:50
Quadrant I
19:04
Quadrant II
19:21
Quadrant III
20:04
Quadrant IV
20:20
Three Dimensions: Space
21:02
Create Ordered Triplets
21:09
Visually Represent This
21:19
Three-Dimension = Space = ℝ³
21:47
Higher Dimensions
22:24
If We Have n Dimensions, We Call It n-Dimensional Space or ℝ to the nth Power
22:31
We Can Represent Places In This n-Dimensional Space As Ordered Groupings of n Numbers
22:41
Hard to Visualize Higher Dimensional Spaces
23:18
Example 1
25:07
Example 2
26:10
Example 3
28:58
Example 4
31:05
Midpoints, Distance, the Pythagorean Theorem, & Slope

48m 43s

Intro
0:00
Introduction
0:07
Midpoint: One Dimension
2:09
Example of Something More Complex
2:31
Use the Idea of a Middle
3:28
Find the Midpoint of Arbitrary Values a and b
4:17
How They're Equivalent
5:05
Official Midpoint Formula
5:46
Midpoint: Two Dimensions
6:19
The Midpoint Must Occur at the Horizontal Middle and the Vertical Middle
6:38
Arbitrary Pair of Points Example
7:25
Distance: One Dimension
9:26
Absolute Value
10:54
Idea of Forcing Positive
11:06
Distance: One Dimension, Formula
11:47
Distance Between Arbitrary a and b
11:48
Absolute Value Helps When the Distance is Negative
12:41
Distance Formula
12:58
The Pythagorean Theorem
13:24
a²+b²=c²
13:50
Distance: Two Dimensions
14:59
Break Into Horizontal and Vertical Parts and then Use the Pythagorean Theorem
15:16
Distance Between Arbitrary Points (x₁,y₁) and (x₂,y₂)
16:21
Slope
19:30
Slope is the Rate of Change
19:41
m = rise over run
21:27
Slope Between Arbitrary Points (x₁,y₁) and (x₂,y₂)
22:31
Interpreting Slope
24:12
Positive Slope and Negative Slope
25:40
m=1, m=0, m=-1
26:48
Example 1
28:25
Example 2
31:42
Example 3
36:40
Example 4
42:48
Word Problems

56m 31s

Intro
0:00
Introduction
0:05
What is a Word Problem?
0:45
Describes Any Problem That Primarily Gets Its Ideas Across With Words Instead of Math Symbols
0:48
Requires Us to Think
1:32
Why Are They So Hard?
2:11
Reason 1: No Simple Formula to Solve Them
2:16
Reason 2: Harder to Teach Word Problems
2:47
You Can Learn How to Do Them!
3:51
Grades
7:57
'But I'm Never Going to Use This In Real Life'
9:46
Solving Word Problems
12:58
First: Understand the Problem
13:37
Second: What Are You Looking For?
14:33
Third: Set Up Relationships
16:21
Fourth: Solve It!
17:48
Summary of Method
19:04
Examples on Things Other Than Math
20:21
Math-Specific Method: What You Need Now
25:30
Understand What the Problem is Talking About
25:37
Set Up and Name Any Variables You Need to Know
25:56
Set Up Equations Connecting Those Variables to the Information in the Problem Statement
26:02
Use the Equations to Solve for an Answer
26:14
Tip
26:58
Draw Pictures
27:22
Breaking Into Pieces
28:28
Try Out Hypothetical Numbers
29:52
Student Logic
31:27
Jump In!
32:40
Example 1
34:03
Example 2
39:15
Example 3
44:22
Example 4
50:24
II. Functions
Idea of a Function

39m 54s

Intro
0:00
Introduction
0:04
What is a Function?
1:06
A Visual Example and Non-Example
1:30
Function Notation
3:47
f(x)
4:05
Express What Sets the Function Acts On
5:45
Metaphors for a Function
6:17
Transformation
6:28
Map
7:17
Machine
8:56
Same Input Always Gives Same Output
10:01
If We Put the Same Input Into a Function, It Will Always Produce the Same Output
10:11
Example of Something That is Not a Function
11:10
A Non-Numerical Example
12:10
The Functions We Will Use
15:05
Unless Told Otherwise, We Will Assume Every Function Takes in Real Numbers and Outputs Real Numbers
15:11
Usually Told the Rule of a Given Function
15:27
How To Use a Function
16:18
Apply the Rule to Whatever Our Input Value Is
16:28
Make Sure to Wrap Your Substitutions in Parentheses
17:09
Functions and Tables
17:36
Table of Values, Sometimes Called a T-Table
17:46
Example
17:56
Domain: What Goes In
18:55
The Domain is the Set of all Inputs That the Function Can Accept
18:56
Example
19:40
Range: What Comes Out
21:27
The Range is the Set of All Possible Outputs a Function Can Assign
21:34
Example
21:49
Another Example Would Be Our Initial Function From Earlier in This Lesson
22:29
Example 1
23:45
Example 2
25:22
Example 3
27:27
Example 4
29:23
Example 5
33:33
Graphs

58m 26s

Intro
0:00
Introduction
0:04
How to Interpret Graphs
1:17
Input / Independent Variable
1:47
Output / Dependent Variable
2:00
Graph as Input ⇒ Output
2:23
One Way to Think of a Graph: See What Happened to Various Inputs
2:25
Example
2:47
Graph as Location of Solution
4:20
A Way to See Solutions
4:36
Example
5:20
Which Way Should We Interpret?
7:13
Easiest to Think In Terms of How Inputs Are Mapped to Outputs
7:20
Sometimes It's Easier to Think In Terms of Solutions
8:39
Pay Attention to Axes
9:50
Axes Tell Where the Graph Is and What Scale It Has
10:09
Often, The Axes Will Be Square
10:14
Example
12:06
Arrows or No Arrows?
16:07
Will Not Use Arrows at the End of Our Graphs
17:13
Graph Stops Because It Hits the Edge of the Graphing Axes, Not Because the Function Stops
17:18
How to Graph
19:47
Plot Points
20:07
Connect with Curves
21:09
If You Connect with Straight Lines
21:44
Graphs of Functions are Smooth
22:21
More Points ⇒ More Accurate
23:38
Vertical Line Test
27:44
If a Vertical Line Could Intersect More Than One Point On a Graph, It Can Not Be the Graph of a Function
28:41
Every Point on a Graph Tells Us Where the x-Value Below is Mapped
30:07
Domain in Graphs
31:37
The Domain is the Set of All Inputs That a Function Can Accept
31:44
Be Aware That Our Function Probably Continues Past the Edge of Our 'Viewing Window'
33:19
Range in Graphs
33:53
Graphing Calculators: Check the Appendix!
36:55
Example 1
38:37
Example 2
45:19
Example 3
50:41
Example 4
53:28
Example 5
55:50
Properties of Functions

48m 49s

Intro
0:00
Introduction
0:05
Increasing Decreasing Constant
0:43
Looking at a Specific Graph
1:15
Increasing Interval
2:39
Constant Function
4:15
Decreasing Interval
5:10
Find Intervals by Looking at the Graph
5:32
Intervals Show x-values; Write in Parentheses
6:39
Maximum and Minimums
8:48
Relative (Local) Max/Min
10:20
Formal Definition of Relative Maximum
12:44
Formal Definition of Relative Minimum
13:05
Max/Min, More Terms
14:18
Definition of Extrema
15:01
Average Rate of Change
16:11
Drawing a Line for the Average Rate
16:48
Using the Slope of the Secant Line
17:36
Slope in Function Notation
18:45
Zeros/Roots/x-intercepts
19:45
What Zeros in a Function Mean
20:25
Even Functions
22:30
Odd Functions
24:36
Even/Odd Functions and Graphs
26:28
Example of an Even Function
27:12
Example of an Odd Function
28:03
Example 1
29:35
Example 2
33:07
Example 3
40:32
Example 4
42:34
Function Petting Zoo

29m 20s

Intro
0:00
Introduction
0:04
Don't Forget that Axes Matter!
1:44
The Constant Function
2:40
The Identity Function
3:44
The Square Function
4:40
The Cube Function
5:44
The Square Root Function
6:51
The Reciprocal Function
8:11
The Absolute Value Function
10:19
The Trigonometric Functions
11:56
f(x)=sin(x)
12:12
f(x)=cos(x)
12:24
Alternate Axes
12:40
The Exponential and Logarithmic Functions
13:35
Exponential Functions
13:44
Logarithmic Functions
14:24
Alternating Axes
15:17
Transformations and Compositions
16:08
Example 1
17:52
Example 2
18:33
Example 3
20:24
Example 4
26:07
Transformation of Functions

48m 35s

Intro
0:00
Introduction
0:04
Vertical Shift
1:12
Graphical Example
1:21
A Further Explanation
2:16
Vertical Stretch/Shrink
3:34
Graph Shrinks
3:46
Graph Stretches
3:51
A Further Explanation
5:07
Horizontal Shift
6:49
Moving the Graph to the Right
7:28
Moving the Graph to the Left
8:12
A Further Explanation
8:19
Understanding Movement on the x-axis
8:38
Horizontal Stretch/Shrink
12:59
Shrinking the Graph
13:40
Stretching the Graph
13:48
A Further Explanation
13:55
Understanding Stretches from the x-axis
14:12
Vertical Flip (aka Mirror)
16:55
Example Graph
17:07
Multiplying the Vertical Component by -1
17:18
Horizontal Flip (aka Mirror)
18:43
Example Graph
19:01
Multiplying the Horizontal Component by -1
19:54
Summary of Transformations
22:11
Stacking Transformations
24:46
Order Matters
25:20
Transformation Example
25:52
Example 1
29:21
Example 2
34:44
Example 3
38:10
Example 4
43:46
Composite Functions

33m 24s

Intro
0:00
Introduction
0:04
Arithmetic Combinations
0:40
Basic Operations
1:20
Definition of the Four Arithmetic Combinations
1:40
Composite Functions
2:53
The Function as a Machine
3:32
Function Compositions as Multiple Machines
3:59
Notation for Composite Functions
4:46
Two Formats
6:02
Another Visual Interpretation
7:17
How to Use Composite Functions
8:21
Example of on Function acting on Another
9:17
Example 1
11:03
Example 2
15:27
Example 3
21:11
Example 4
27:06
Piecewise Functions

51m 42s

Intro
0:00
Introduction
0:04
Analogies to a Piecewise Function
1:16
Different Potatoes
1:41
Factory Production
2:27
Notations for Piecewise Functions
3:39
Notation Examples from Analogies
6:11
Example of a Piecewise (with Table)
7:24
Example of a Non-Numerical Piecewise
11:35
Graphing Piecewise Functions
14:15
Graphing Piecewise Functions, Example
16:26
Continuous Functions
16:57
Statements of Continuity
19:30
Example of Continuous and Non-Continuous Graphs
20:05
Interesting Functions: the Step Function
22:00
Notation for the Step Function
22:40
How the Step Function Works
22:56
Graph of the Step Function
25:30
Example 1
26:22
Example 2
28:49
Example 3
36:50
Example 4
46:11
Inverse Functions

49m 37s

Intro
0:00
Introduction
0:04
Analogy by picture
1:10
How to Denote the inverse
1:40
What Comes out of the Inverse
1:52
Requirement for Reversing
2:02
The Basketball Factory
2:12
The Importance of Information
2:45
One-to-One
4:04
Requirement for Reversibility
4:21
When a Function has an Inverse
4:43
One-to-One
5:13
Not One-to-One
5:50
Not a Function
6:19
Horizontal Line Test
7:01
How to the test Works
7:12
One-to-One
8:12
Not One-to-One
8:45
Definition: Inverse Function
9:12
Formal Definition
9:21
Caution to Students
10:02
Domain and Range
11:12
Finding the Range of the Function Inverse
11:56
Finding the Domain of the Function Inverse
12:11
Inverse of an Inverse
13:09
Its just x!
13:26
Proof
14:03
Graphical Interpretation
17:07
Horizontal Line Test
17:20
Graph of the Inverse
18:04
Swapping Inputs and Outputs to Draw Inverses
19:02
How to Find the Inverse
21:03
What We Are Looking For
21:21
Reversing the Function
21:38
A Method to Find Inverses
22:33
Check Function is One-to-One
23:04
Swap f(x) for y
23:25
Interchange x and y
23:41
Solve for y
24:12
Replace y with the inverse
24:40
Some Comments
25:01
Keeping Step 2 and 3 Straight
25:44
Switching to Inverse
26:12
Checking Inverses
28:52
How to Check an Inverse
29:06
Quick Example of How to Check
29:56
Example 1
31:48
Example 2
34:56
Example 3
39:29
Example 4
46:19
Variation Direct and Inverse

28m 49s

Intro
0:00
Introduction
0:06
Direct Variation
1:14
Same Direction
1:21
Common Example: Groceries
1:56
Different Ways to Say that Two Things Vary Directly
2:28
Basic Equation for Direct Variation
2:55
Inverse Variation
3:40
Opposite Direction
3:50
Common Example: Gravity
4:53
Different Ways to Say that Two Things Vary Indirectly
5:48
Basic Equation for Indirect Variation
6:33
Joint Variation
7:27
Equation for Joint Variation
7:53
Explanation of the Constant
8:48
Combined Variation
9:35
Gas Law as a Combination
9:44
Single Constant
10:33
Example 1
10:49
Example 2
13:34
Example 3
15:39
Example 4
19:48
III. Polynomials
Intro to Polynomials

38m 41s

Intro
0:00
Introduction
0:04
Definition of a Polynomial
1:04
Starting Integer
2:06
Structure of a Polynomial
2:49
The a Constants
3:34
Polynomial Function
5:13
Polynomial Equation
5:23
Polynomials with Different Variables
5:36
Degree
6:23
Informal Definition
6:31
Find the Largest Exponent Variable
6:44
Quick Examples
7:36
Special Names for Polynomials
8:59
Based on the Degree
9:23
Based on the Number of Terms
10:12
Distributive Property (aka 'FOIL')
11:37
Basic Distributive Property
12:21
Distributing Two Binomials
12:55
Longer Parentheses
15:12
Reverse: Factoring
17:26
Long-Term Behavior of Polynomials
17:48
Examples
18:13
Controlling Term--Term with the Largest Exponent
19:33
Positive and Negative Coefficients on the Controlling Term
20:21
Leading Coefficient Test
22:07
Even Degree, Positive Coefficient
22:13
Even Degree, Negative Coefficient
22:39
Odd Degree, Positive Coefficient
23:09
Odd Degree, Negative Coefficient
23:27
Example 1
25:11
Example 2
27:16
Example 3
31:16
Example 4
34:41
Roots (Zeros) of Polynomials

41m 7s

Intro
0:00
Introduction
0:05
Roots in Graphs
1:17
The x-intercepts
1:33
How to Remember What 'Roots' Are
1:50
Naïve Attempts
2:31
Isolating Variables
2:45
Failures of Isolating Variables
3:30
Missing Solutions
4:59
Factoring: How to Find Roots
6:28
How Factoring Works
6:36
Why Factoring Works
7:20
Steps to Finding Polynomial Roots
9:21
Factoring: How to Find Roots CAUTION
10:08
Factoring is Not Easy
11:32
Factoring Quadratics
13:08
Quadratic Trinomials
13:21
Form of Factored Binomials
13:38
Factoring Examples
14:40
Factoring Quadratics, Check Your Work
16:58
Factoring Higher Degree Polynomials
18:19
Factoring a Cubic
18:32
Factoring a Quadratic
19:04
Factoring: Roots Imply Factors
19:54
Where a Root is, A Factor Is
20:01
How to Use Known Roots to Make Factoring Easier
20:35
Not all Polynomials Can be Factored
22:30
Irreducible Polynomials
23:27
Complex Numbers Help
23:55
Max Number of Roots/Factors
24:57
Limit to Number of Roots Equal to the Degree
25:18
Why there is a Limit
25:25
Max Number of Peaks/Valleys
26:39
Shape Information from Degree
26:46
Example Graph
26:54
Max, But Not Required
28:00
Example 1
28:37
Example 2
31:21
Example 3
36:12
Example 4
38:40
Completing the Square and the Quadratic Formula

39m 43s

Intro
0:00
Introduction
0:05
Square Roots and Equations
0:51
Taking the Square Root to Find the Value of x
0:55
Getting the Positive and Negative Answers
1:05
Completing the Square: Motivation
2:04
Polynomials that are Easy to Solve
2:20
Making Complex Polynomials Easy to Solve
3:03
Steps to Completing the Square
4:30
Completing the Square: Method
7:22
Move C over
7:35
Divide by A
7:44
Find r
7:59
Add to Both Sides to Complete the Square
8:49
Solving Quadratics with Ease
9:56
The Quadratic Formula
11:38
Derivation
11:43
Final Form
12:23
Follow Format to Use Formula
13:38
How Many Roots?
14:53
The Discriminant
15:47
What the Discriminant Tells Us: How Many Roots
15:58
How the Discriminant Works
16:30
Example 1: Complete the Square
18:24
Example 2: Solve the Quadratic
22:00
Example 3: Solve for Zeroes
25:28
Example 4: Using the Quadratic Formula
30:52
Properties of Quadratic Functions

45m 34s

Intro
0:00
Introduction
0:05
Parabolas
0:35
Examples of Different Parabolas
1:06
Axis of Symmetry and Vertex
1:28
Drawing an Axis of Symmetry
1:51
Placing the Vertex
2:28
Looking at the Axis of Symmetry and Vertex for other Parabolas
3:09
Transformations
4:18
Reviewing Transformation Rules
6:28
Note the Different Horizontal Shift Form
7:45
An Alternate Form to Quadratics
8:54
The Constants: k, h, a
9:05
Transformations Formed
10:01
Analyzing Different Parabolas
10:10
Switching Forms by Completing the Square
11:43
Vertex of a Parabola
16:30
Vertex at (h, k)
16:47
Vertex in Terms of a, b, and c Coefficients
17:28
Minimum/Maximum at Vertex
18:19
When a is Positive
18:25
When a is Negative
18:52
Axis of Symmetry
19:54
Incredibly Minor Note on Grammar
20:52
Example 1
21:48
Example 2
26:35
Example 3
28:55
Example 4
31:40
Intermediate Value Theorem and Polynomial Division

46m 8s

Intro
0:00
Introduction
0:05
Reminder: Roots Imply Factors
1:32
The Intermediate Value Theorem
3:41
The Basis: U between a and b
4:11
U is on the Function
4:52
Intermediate Value Theorem, Proof Sketch
5:51
If Not True, the Graph Would Have to Jump
5:58
But Graph is Defined as Continuous
6:43
Finding Roots with the Intermediate Value Theorem
7:01
Picking a and b to be of Different Signs
7:10
Must Be at Least One Root
7:46
Dividing a Polynomial
8:16
Using Roots and Division to Factor
8:38
Long Division Refresher
9:08
The Division Algorithm
12:18
How It Works to Divide Polynomials
12:37
The Parts of the Equation
13:24
Rewriting the Equation
14:47
Polynomial Long Division
16:20
Polynomial Long Division In Action
16:29
One Step at a Time
20:51
Synthetic Division
22:46
Setup
23:11
Synthetic Division, Example
24:44
Which Method Should We Use
26:39
Advantages of Synthetic Method
26:49
Advantages of Long Division
27:13
Example 1
29:24
Example 2
31:27
Example 3
36:22
Example 4
40:55
Complex Numbers

45m 36s

Intro
0:00
Introduction
0:04
A Wacky Idea
1:02
The Definition of the Imaginary Number
1:22
How it Helps Solve Equations
2:20
Square Roots and Imaginary Numbers
3:15
Complex Numbers
5:00
Real Part and Imaginary Part
5:20
When Two Complex Numbers are Equal
6:10
Addition and Subtraction
6:40
Deal with Real and Imaginary Parts Separately
7:36
Two Quick Examples
7:54
Multiplication
9:07
FOIL Expansion
9:14
Note What Happens to the Square of the Imaginary Number
9:41
Two Quick Examples
10:22
Division
11:27
Complex Conjugates
13:37
Getting Rid of i
14:08
How to Denote the Conjugate
14:48
Division through Complex Conjugates
16:11
Multiply by the Conjugate of the Denominator
16:28
Example
17:46
Factoring So-Called 'Irreducible' Quadratics
19:24
Revisiting the Quadratic Formula
20:12
Conjugate Pairs
20:37
But Are the Complex Numbers 'Real'?
21:27
What Makes a Number Legitimate
25:38
Where Complex Numbers are Used
27:20
Still, We Won't See Much of C
29:05
Example 1
30:30
Example 2
33:15
Example 3
38:12
Example 4
42:07
Fundamental Theorem of Algebra

19m 9s

Intro
0:00
Introduction
0:05
Idea: Hidden Roots
1:16
Roots in Complex Form
1:42
All Polynomials Have Roots
2:08
Fundamental Theorem of Algebra
2:21
Where Are All the Imaginary Roots, Then?
3:17
All Roots are Complex
3:45
Real Numbers are a Subset of Complex Numbers
3:59
The n Roots Theorem
5:01
For Any Polynomial, Its Degree is Equal to the Number of Roots
5:11
Equivalent Statement
5:24
Comments: Multiplicity
6:29
Non-Distinct Roots
6:59
Denoting Multiplicity
7:20
Comments: Complex Numbers Necessary
7:41
Comments: Complex Coefficients Allowed
8:55
Comments: Existence Theorem
9:59
Proof Sketch of n Roots Theorem
10:45
First Root
11:36
Second Root
13:23
Continuation to Find all Roots
16:00
IV. Rational Functions
Rational Functions and Vertical Asymptotes

33m 22s

Intro
0:00
Introduction
0:05
Definition of a Rational Function
1:20
Examples of Rational Functions
2:30
Why They are Called 'Rational'
2:47
Domain of a Rational Function
3:15
Undefined at Denominator Zeros
3:25
Otherwise all Reals
4:16
Investigating a Fundamental Function
4:50
The Domain of the Function
5:04
What Occurs at the Zeroes of the Denominator
5:20
Idea of a Vertical Asymptote
6:23
What's Going On?
6:58
Approaching x=0 from the left
7:32
Approaching x=0 from the right
8:34
Dividing by Very Small Numbers Results in Very Large Numbers
9:31
Definition of a Vertical Asymptote
10:05
Vertical Asymptotes and Graphs
11:15
Drawing Asymptotes by Using a Dashed Line
11:27
The Graph Can Never Touch Its Undefined Point
12:00
Not All Zeros Give Asymptotes
13:02
Special Cases: When Numerator and Denominator Go to Zero at the Same Time
14:58
Cancel out Common Factors
15:49
How to Find Vertical Asymptotes
16:10
Figure out What Values Are Not in the Domain of x
16:24
Determine if the Numerator and Denominator Share Common Factors and Cancel
16:45
Find Denominator Roots
17:33
Note if Asymptote Approaches Negative or Positive Infinity
18:06
Example 1
18:57
Example 2
21:26
Example 3
23:04
Example 4
30:01
Horizontal Asymptotes

34m 16s

Intro
0:00
Introduction
0:05
Investigating a Fundamental Function
0:53
What Happens as x Grows Large
1:00
Different View
1:12
Idea of a Horizontal Asymptote
1:36
What's Going On?
2:24
What Happens as x Grows to a Large Negative Number
2:49
What Happens as x Grows to a Large Number
3:30
Dividing by Very Large Numbers Results in Very Small Numbers
3:52
Example Function
4:41
Definition of a Vertical Asymptote
8:09
Expanding the Idea
9:03
What's Going On?
9:48
What Happens to the Function in the Long Run?
9:51
Rewriting the Function
10:13
Definition of a Slant Asymptote
12:09
Symbolical Definition
12:30
Informal Definition
12:45
Beyond Slant Asymptotes
13:03
Not Going Beyond Slant Asymptotes
14:39
Horizontal/Slant Asymptotes and Graphs
15:43
How to Find Horizontal and Slant Asymptotes
16:52
How to Find Horizontal Asymptotes
17:12
Expand the Given Polynomials
17:18
Compare the Degrees of the Numerator and Denominator
17:40
How to Find Slant Asymptotes
20:05
Slant Asymptotes Exist When n+m=1
20:08
Use Polynomial Division
20:24
Example 1
24:32
Example 2
25:53
Example 3
26:55
Example 4
29:22
Graphing Asymptotes in a Nutshell

49m 7s

Intro
0:00
Introduction
0:05
A Process for Graphing
1:22
1. Factor Numerator and Denominator
1:50
2. Find Domain
2:53
3. Simplifying the Function
3:59
4. Find Vertical Asymptotes
4:59
5. Find Horizontal/Slant Asymptotes
5:24
6. Find Intercepts
7:35
7. Draw Graph (Find Points as Necessary)
9:21
Draw Graph Example
11:21
Vertical Asymptote
11:41
Horizontal Asymptote
11:50
Other Graphing
12:16
Test Intervals
15:08
Example 1
17:57
Example 2
23:01
Example 3
29:02
Example 4
33:37
Partial Fractions

44m 56s

Intro
0:00
Introduction: Idea
0:04
Introduction: Prerequisites and Uses
1:57
Proper vs. Improper Polynomial Fractions
3:11
Possible Things in the Denominator
4:38
Linear Factors
6:16
Example of Linear Factors
7:03
Multiple Linear Factors
7:48
Irreducible Quadratic Factors
8:25
Example of Quadratic Factors
9:26
Multiple Quadratic Factors
9:49
Mixing Factor Types
10:28
Figuring Out the Numerator
11:10
How to Solve for the Constants
11:30
Quick Example
11:40
Example 1
14:29
Example 2
18:35
Example 3
20:33
Example 4
28:51
V. Exponential & Logarithmic Functions
Understanding Exponents

35m 17s

Intro
0:00
Introduction
0:05
Fundamental Idea
1:46
Expanding the Idea
2:28
Multiplication of the Same Base
2:40
Exponents acting on Exponents
3:45
Different Bases with the Same Exponent
4:31
To the Zero
5:35
To the First
5:45
Fundamental Rule with the Zero Power
6:35
To the Negative
7:45
Any Number to a Negative Power
8:14
A Fraction to a Negative Power
9:58
Division with Exponential Terms
10:41
To the Fraction
11:33
Square Root
11:58
Any Root
12:59
Summary of Rules
14:38
To the Irrational
17:21
Example 1
20:34
Example 2
23:42
Example 3
27:44
Example 4
31:44
Example 5
33:15
Exponential Functions

47m 4s

Intro
0:00
Introduction
0:05
Definition of an Exponential Function
0:48
Definition of the Base
1:02
Restrictions on the Base
1:16
Computing Exponential Functions
2:29
Harder Computations
3:10
When to Use a Calculator
3:21
Graphing Exponential Functions: a>1
6:02
Three Examples
6:13
What to Notice on the Graph
7:44
A Story
8:27
Story Diagram
9:15
Increasing Exponentials
11:29
Story Morals
14:40
Application: Compound Interest
15:15
Compounding Year after Year
16:01
Function for Compounding Interest
16:51
A Special Number: e
20:55
Expression for e
21:28
Where e stabilizes
21:55
Application: Continuously Compounded Interest
24:07
Equation for Continuous Compounding
24:22
Exponential Decay 0<a<1
25:50
Three Examples
26:11
Why they 'lose' value
26:54
Example 1
27:47
Example 2
33:11
Example 3
36:34
Example 4
41:28
Introduction to Logarithms

40m 31s

Intro
0:00
Introduction
0:04
Definition of a Logarithm, Base 2
0:51
Log 2 Defined
0:55
Examples
2:28
Definition of a Logarithm, General
3:23
Examples of Logarithms
5:15
Problems with Unusual Bases
7:38
Shorthand Notation: ln and log
9:44
base e as ln
10:01
base 10 as log
10:34
Calculating Logarithms
11:01
using a calculator
11:34
issues with other bases
11:58
Graphs of Logarithms
13:21
Three Examples
13:29
Slow Growth
15:19
Logarithms as Inverse of Exponentiation
16:02
Using Base 2
16:05
General Case
17:10
Looking More Closely at Logarithm Graphs
19:16
The Domain of Logarithms
20:41
Thinking about Logs like Inverses
21:08
The Alternate
24:00
Example 1
25:59
Example 2
30:03
Example 3
32:49
Example 4
37:34
Properties of Logarithms

42m 33s

Intro
0:00
Introduction
0:04
Basic Properties
1:12
Inverse--log(exp)
1:43
A Key Idea
2:44
What We Get through Exponentiation
3:18
B Always Exists
4:50
Inverse--exp(log)
5:53
Logarithm of a Power
7:44
Logarithm of a Product
10:07
Logarithm of a Quotient
13:48
Caution! There Is No Rule for loga(M+N)
16:12
Summary of Properties
17:42
Change of Base--Motivation
20:17
No Calculator Button
20:59
A Specific Example
21:45
Simplifying
23:45
Change of Base--Formula
24:14
Example 1
25:47
Example 2
29:08
Example 3
31:14
Example 4
34:13
Solving Exponential and Logarithmic Equations

34m 10s

Intro
0:00
Introduction
0:05
One to One Property
1:09
Exponential
1:26
Logarithmic
1:44
Specific Considerations
2:02
One-to-One Property
3:30
Solving by One-to-One
4:11
Inverse Property
6:09
Solving by Inverses
7:25
Dealing with Equations
7:50
Example of Taking an Exponent or Logarithm of an Equation
9:07
A Useful Property
11:57
Bring Down Exponents
12:01
Try to Simplify
13:20
Extraneous Solutions
13:45
Example 1
16:37
Example 2
19:39
Example 3
21:37
Example 4
26:45
Example 5
29:37
Application of Exponential and Logarithmic Functions

48m 46s

Intro
0:00
Introduction
0:06
Applications of Exponential Functions
1:07
A Secret!
2:17
Natural Exponential Growth Model
3:07
Figure out r
3:34
A Secret!--Why Does It Work?
4:44
e to the r Morphs
4:57
Example
5:06
Applications of Logarithmic Functions
8:32
Examples
8:43
What Logarithms are Useful For
9:53
Example 1
11:29
Example 2
15:30
Example 3
26:22
Example 4
32:05
Example 5
39:19
VI. Trigonometric Functions
Angles

39m 5s

Intro
0:00
Degrees
0:22
Circle is 360 Degrees
0:48
Splitting a Circle
1:13
Radians
2:08
Circle is 2 Pi Radians
2:31
One Radian
2:52
Half-Circle and Right Angle
4:00
Converting Between Degrees and Radians
6:24
Formulas for Degrees and Radians
6:52
Coterminal, Complementary, Supplementary Angles
7:23
Coterminal Angles
7:30
Complementary Angles
9:40
Supplementary Angles
10:08
Example 1: Dividing a Circle
10:38
Example 2: Converting Between Degrees and Radians
11:56
Example 3: Quadrants and Coterminal Angles
14:18
Extra Example 1: Common Angle Conversions
-1
Extra Example 2: Quadrants and Coterminal Angles
-2
Sine and Cosine Functions

43m 16s

Intro
0:00
Sine and Cosine
0:15
Unit Circle
0:22
Coordinates on Unit Circle
1:03
Right Triangles
1:52
Adjacent, Opposite, Hypotenuse
2:25
Master Right Triangle Formula: SOHCAHTOA
2:48
Odd Functions, Even Functions
4:40
Example: Odd Function
4:56
Example: Even Function
7:30
Example 1: Sine and Cosine
10:27
Example 2: Graphing Sine and Cosine Functions
14:39
Example 3: Right Triangle
21:40
Example 4: Odd, Even, or Neither
26:01
Extra Example 1: Right Triangle
-1
Extra Example 2: Graphing Sine and Cosine Functions
-2
Sine and Cosine Values of Special Angles

33m 5s

Intro
0:00
45-45-90 Triangle and 30-60-90 Triangle
0:08
45-45-90 Triangle
0:21
30-60-90 Triangle
2:06
Mnemonic: All Students Take Calculus (ASTC)
5:21
Using the Unit Circle
5:59
New Angles
6:21
Other Quadrants
9:43
Mnemonic: All Students Take Calculus
10:13
Example 1: Convert, Quadrant, Sine/Cosine
13:11
Example 2: Convert, Quadrant, Sine/Cosine
16:48
Example 3: All Angles and Quadrants
20:21
Extra Example 1: Convert, Quadrant, Sine/Cosine
-1
Extra Example 2: All Angles and Quadrants
-2
Modified Sine Waves: Asin(Bx+C)+D and Acos(Bx+C)+D

52m 3s

Intro
0:00
Amplitude and Period of a Sine Wave
0:38
Sine Wave Graph
0:58
Amplitude: Distance from Middle to Peak
1:18
Peak: Distance from Peak to Peak
2:41
Phase Shift and Vertical Shift
4:13
Phase Shift: Distance Shifted Horizontally
4:16
Vertical Shift: Distance Shifted Vertically
6:48
Example 1: Amplitude/Period/Phase and Vertical Shift
8:04
Example 2: Amplitude/Period/Phase and Vertical Shift
17:39
Example 3: Find Sine Wave Given Attributes
25:23
Extra Example 1: Amplitude/Period/Phase and Vertical Shift
-1
Extra Example 2: Find Cosine Wave Given Attributes
-2
Tangent and Cotangent Functions

36m 4s

Intro
0:00
Tangent and Cotangent Definitions
0:21
Tangent Definition
0:25
Cotangent Definition
0:47
Master Formula: SOHCAHTOA
1:01
Mnemonic
1:16
Tangent and Cotangent Values
2:29
Remember Common Values of Sine and Cosine
2:46
90 Degrees Undefined
4:36
Slope and Menmonic: ASTC
5:47
Uses of Tangent
5:54
Example: Tangent of Angle is Slope
6:09
Sign of Tangent in Quadrants
7:49
Example 1: Graph Tangent and Cotangent Functions
10:42
Example 2: Tangent and Cotangent of Angles
16:09
Example 3: Odd, Even, or Neither
18:56
Extra Example 1: Tangent and Cotangent of Angles
-1
Extra Example 2: Tangent and Cotangent of Angles
-2
Secant and Cosecant Functions

27m 18s

Intro
0:00
Secant and Cosecant Definitions
0:17
Secant Definition
0:18
Cosecant Definition
0:33
Example 1: Graph Secant Function
0:48
Example 2: Values of Secant and Cosecant
6:49
Example 3: Odd, Even, or Neither
12:49
Extra Example 1: Graph of Cosecant Function
-1
Extra Example 2: Values of Secant and Cosecant
-2
Inverse Trigonometric Functions

32m 58s

Intro
0:00
Arcsine Function
0:24
Restrictions between -1 and 1
0:43
Arcsine Notation
1:26
Arccosine Function
3:07
Restrictions between -1 and 1
3:36
Cosine Notation
3:53
Arctangent Function
4:30
Between -Pi/2 and Pi/2
4:44
Tangent Notation
5:02
Example 1: Domain/Range/Graph of Arcsine
5:45
Example 2: Arcsin/Arccos/Arctan Values
10:46
Example 3: Domain/Range/Graph of Arctangent
17:14
Extra Example 1: Domain/Range/Graph of Arccosine
-1
Extra Example 2: Arcsin/Arccos/Arctan Values
-2
Computations of Inverse Trigonometric Functions

31m 8s

Intro
0:00
Inverse Trigonometric Function Domains and Ranges
0:31
Arcsine
0:41
Arccosine
1:14
Arctangent
1:41
Example 1: Arcsines of Common Values
2:44
Example 2: Odd, Even, or Neither
5:57
Example 3: Arccosines of Common Values
12:24
Extra Example 1: Arctangents of Common Values
-1
Extra Example 2: Arcsin/Arccos/Arctan Values
-2
VII. Trigonometric Identities
Pythagorean Identity

19m 11s

Intro
0:00
Pythagorean Identity
0:17
Pythagorean Triangle
0:27
Pythagorean Identity
0:45
Example 1: Use Pythagorean Theorem to Prove Pythagorean Identity
1:14
Example 2: Find Angle Given Cosine and Quadrant
4:18
Example 3: Verify Trigonometric Identity
8:00
Extra Example 1: Use Pythagorean Identity to Prove Pythagorean Theorem
-1
Extra Example 2: Find Angle Given Cosine and Quadrant
-2
Identity Tan(squared)x+1=Sec(squared)x

23m 16s

Intro
0:00
Main Formulas
0:19
Companion to Pythagorean Identity
0:27
For Cotangents and Cosecants
0:52
How to Remember
0:58
Example 1: Prove the Identity
1:40
Example 2: Given Tan Find Sec
3:42
Example 3: Prove the Identity
7:45
Extra Example 1: Prove the Identity
-1
Extra Example 2: Given Sec Find Tan
-2
Addition and Subtraction Formulas

52m 52s

Intro
0:00
Addition and Subtraction Formulas
0:09
How to Remember
0:48
Cofunction Identities
1:31
How to Remember Graphically
1:44
Where to Use Cofunction Identities
2:52
Example 1: Derive the Formula for cos(A-B)
3:08
Example 2: Use Addition and Subtraction Formulas
16:03
Example 3: Use Addition and Subtraction Formulas to Prove Identity
25:11
Extra Example 1: Use cos(A-B) and Cofunction Identities
-1
Extra Example 2: Convert to Radians and use Formulas
-2
Double Angle Formulas

29m 5s

Intro
0:00
Main Formula
0:07
How to Remember from Addition Formula
0:18
Two Other Forms
1:35
Example 1: Find Sine and Cosine of Angle using Double Angle
3:16
Example 2: Prove Trigonometric Identity using Double Angle
9:37
Example 3: Use Addition and Subtraction Formulas
12:38
Extra Example 1: Find Sine and Cosine of Angle using Double Angle
-1
Extra Example 2: Prove Trigonometric Identity using Double Angle
-2
Half-Angle Formulas

43m 55s

Intro
0:00
Main Formulas
0:09
Confusing Part
0:34
Example 1: Find Sine and Cosine of Angle using Half-Angle
0:54
Example 2: Prove Trigonometric Identity using Half-Angle
11:51
Example 3: Prove the Half-Angle Formula for Tangents
18:39
Extra Example 1: Find Sine and Cosine of Angle using Half-Angle
-1
Extra Example 2: Prove Trigonometric Identity using Half-Angle
-2
VIII. Applications of Trigonometry
Trigonometry in Right Angles

25m 43s

Intro
0:00
Master Formula for Right Angles
0:11
SOHCAHTOA
0:15
Only for Right Triangles
1:26
Example 1: Find All Angles in a Triangle
2:19
Example 2: Find Lengths of All Sides of Triangle
7:39
Example 3: Find All Angles in a Triangle
11:00
Extra Example 1: Find All Angles in a Triangle
-1
Extra Example 2: Find Lengths of All Sides of Triangle
-2
Law of Sines

56m 40s

Intro
0:00
Law of Sines Formula
0:18
SOHCAHTOA
0:27
Any Triangle
0:59
Graphical Representation
1:25
Solving Triangle Completely
2:37
When to Use Law of Sines
2:55
ASA, SAA, SSA, AAA
2:59
SAS, SSS for Law of Cosines
7:11
Example 1: How Many Triangles Satisfy Conditions, Solve Completely
8:44
Example 2: How Many Triangles Satisfy Conditions, Solve Completely
15:30
Example 3: How Many Triangles Satisfy Conditions, Solve Completely
28:32
Extra Example 1: How Many Triangles Satisfy Conditions, Solve Completely
-1
Extra Example 2: How Many Triangles Satisfy Conditions, Solve Completely
-2
Law of Cosines

49m 5s

Intro
0:00
Law of Cosines Formula
0:23
Graphical Representation
0:34
Relates Sides to Angles
1:00
Any Triangle
1:20
Generalization of Pythagorean Theorem
1:32
When to Use Law of Cosines
2:26
SAS, SSS
2:30
Heron's Formula
4:49
Semiperimeter S
5:11
Example 1: How Many Triangles Satisfy Conditions, Solve Completely
5:53
Example 2: How Many Triangles Satisfy Conditions, Solve Completely
15:19
Example 3: Find Area of a Triangle Given All Side Lengths
26:33
Extra Example 1: How Many Triangles Satisfy Conditions, Solve Completely
-1
Extra Example 2: Length of Third Side and Area of Triangle
-2
Finding the Area of a Triangle

27m 37s

Intro
0:00
Master Right Triangle Formula and Law of Cosines
0:19
SOHCAHTOA
0:27
Law of Cosines
1:23
Heron's Formula
2:22
Semiperimeter S
2:37
Example 1: Area of Triangle with Two Sides and One Angle
3:12
Example 2: Area of Triangle with Three Sides
6:11
Example 3: Area of Triangle with Three Sides, No Heron's Formula
8:50
Extra Example 1: Area of Triangle with Two Sides and One Angle
-1
Extra Example 2: Area of Triangle with Two Sides and One Angle
-2
Word Problems and Applications of Trigonometry

34m 25s

Intro
0:00
Formulas to Remember
0:11
SOHCAHTOA
0:15
Law of Sines
0:55
Law of Cosines
1:48
Heron's Formula
2:46
Example 1: Telephone Pole Height
4:01
Example 2: Bridge Length
7:48
Example 3: Area of Triangular Field
14:20
Extra Example 1: Kite Height
-1
Extra Example 2: Roads to a Town
-2
IX. Systems of Equations and Inequalities
Systems of Linear Equations

55m 40s

Intro
0:00
Introduction
0:04
Graphs as Location of 'True'
1:49
All Locations that Make the Function True
2:25
Understand the Relationship Between Solutions and the Graph
3:43
Systems as Graphs
4:07
Equations as Lines
4:20
Intersection Point
5:19
Three Possibilities for Solutions
6:17
Independent
6:24
Inconsistent
6:36
Dependent
7:06
Solving by Substitution
8:37
Solve for One Variable
9:07
Substitute into the Second Equation
9:34
Solve for Both Variables
10:12
What If a System is Inconsistent or Dependent?
11:08
No Solutions
11:25
Infinite Solutions
12:30
Solving by Elimination
13:56
Example
14:22
Determining the Number of Solutions
16:30
Why Elimination Makes Sense
17:25
Solving by Graphing Calculator
19:59
Systems with More than Two Variables
23:22
Example 1
25:49
Example 2
30:22
Example 3
34:11
Example 4
38:55
Example 5
46:01
(Non-) Example 6
53:37
Systems of Linear Inequalities

1h 13s

Intro
0:00
Introduction
0:04
Inequality Refresher-Solutions
0:46
Equation Solutions vs. Inequality Solutions
1:02
Essentially a Wide Variety of Answers
1:35
Refresher--Negative Multiplication Flips
1:43
Refresher--Negative Flips: Why?
3:19
Multiplication by a Negative
3:43
The Relationship Flips
3:55
Refresher--Stick to Basic Operations
4:34
Linear Equations in Two Variables
6:50
Graphing Linear Inequalities
8:28
Why It Includes a Whole Section
8:43
How to Show The Difference Between Strict and Not Strict Inequalities
10:08
Dashed Line--Not Solutions
11:10
Solid Line--Are Solutions
11:24
Test Points for Shading
11:42
Example of Using a Point
12:41
Drawing Shading from the Point
13:14
Graphing a System
14:53
Set of Solutions is the Overlap
15:17
Example
15:22
Solutions are Best Found Through Graphing
18:05
Linear Programming-Idea
19:52
Use a Linear Objective Function
20:15
Variables in Objective Function have Constraints
21:24
Linear Programming-Method
22:09
Rearrange Equations
22:21
Graph
22:49
Critical Solution is at the Vertex of the Overlap
23:40
Try Each Vertice
24:35
Example 1
24:58
Example 2
28:57
Example 3
33:48
Example 4
43:10
Nonlinear Systems

41m 1s

Intro
0:00
Introduction
0:06
Substitution
1:12
Example
1:22
Elimination
3:46
Example
3:56
Elimination is Less Useful for Nonlinear Systems
4:56
Graphing
5:56
Using a Graphing Calculator
6:44
Number of Solutions
8:44
Systems of Nonlinear Inequalities
10:02
Graph Each Inequality
10:06
Dashed and/or Solid
10:18
Shade Appropriately
11:14
Example 1
13:24
Example 2
15:50
Example 3
22:02
Example 4
29:06
Example 4, cont.
33:40
X. Vectors and Matrices
Vectors

1h 9m 31s

Intro
0:00
Introduction
0:10
Magnitude of the Force
0:22
Direction of the Force
0:48
Vector
0:52
Idea of a Vector
1:30
How Vectors are Denoted
2:00
Component Form
3:20
Angle Brackets and Parentheses
3:50
Magnitude/Length
4:26
Denoting the Magnitude of a Vector
5:16
Direction/Angle
7:52
Always Draw a Picture
8:50
Component Form from Magnitude & Angle
10:10
Scaling by Scalars
14:06
Unit Vectors
16:26
Combining Vectors - Algebraically
18:10
Combining Vectors - Geometrically
19:54
Resultant Vector
20:46
Alternate Component Form: i, j
21:16
The Zero Vector
23:18
Properties of Vectors
24:20
No Multiplication (Between Vectors)
28:30
Dot Product
29:40
Motion in a Medium
30:10
Fish in an Aquarium Example
31:38
More Than Two Dimensions
33:12
More Than Two Dimensions - Magnitude
34:18
Example 1
35:26
Example 2
38:10
Example 3
45:48
Example 4
50:40
Example 4, cont.
56:07
Example 5
1:01:32
Dot Product & Cross Product

35m 20s

Intro
0:00
Introduction
0:08
Dot Product - Definition
0:42
Dot Product Results in a Scalar, Not a Vector
2:10
Example in Two Dimensions
2:34
Angle and the Dot Product
2:58
The Dot Product of Two Vectors is Deeply Related to the Angle Between the Two Vectors
2:59
Proof of Dot Product Formula
4:14
Won't Directly Help Us Better Understand Vectors
4:18
Dot Product - Geometric Interpretation
4:58
We Can Interpret the Dot Product as a Measure of How Long and How Parallel Two Vectors Are
7:26
Dot Product - Perpendicular Vectors
8:24
If the Dot Product of Two Vectors is 0, We Know They are Perpendicular to Each Other
8:54
Cross Product - Definition
11:08
Cross Product Only Works in Three Dimensions
11:09
Cross Product - A Mnemonic
12:16
The Determinant of a 3 x 3 Matrix and Standard Unit Vectors
12:17
Cross Product - Geometric Interpretations
14:30
The Right-Hand Rule
15:17
Cross Product - Geometric Interpretations Cont.
17:00
Example 1
18:40
Example 2
22:50
Example 3
24:04
Example 4
26:20
Bonus Round
29:18
Proof: Dot Product Formula
29:24
Proof: Dot Product Formula, cont.
30:38
Matrices

54m 7s

Intro
0:00
Introduction
0:08
Definition of a Matrix
3:02
Size or Dimension
3:58
Square Matrix
4:42
Denoted by Capital Letters
4:56
When are Two Matrices Equal?
5:04
Examples of Matrices
6:44
Rows x Columns
6:46
Talking About Specific Entries
7:48
We Use Capitals to Denote a Matrix and Lower Case to Denotes Its Entries
8:32
Using Entries to Talk About Matrices
10:08
Scalar Multiplication
11:26
Scalar = Real Number
11:34
Example
12:36
Matrix Addition
13:08
Example
14:22
Matrix Multiplication
15:00
Example
18:52
Matrix Multiplication, cont.
19:58
Matrix Multiplication and Order (Size)
25:26
Make Sure Their Orders are Compatible
25:27
Matrix Multiplication is NOT Commutative
28:20
Example
30:08
Special Matrices - Zero Matrix (0)
32:48
Zero Matrix Has 0 for All of its Entries
32:49
Special Matrices - Identity Matrix (I)
34:14
Identity Matrix is a Square Matrix That Has 1 for All Its Entries on the Main Diagonal and 0 for All Other Entries
34:15
Example 1
36:16
Example 2
40:00
Example 3
44:54
Example 4
50:08
Determinants & Inverses of Matrices

47m 12s

Intro
0:00
Introduction
0:06
Not All Matrices Are Invertible
1:30
What Must a Matrix Have to Be Invertible?
2:08
Determinant
2:32
The Determinant is a Real Number Associated With a Square Matrix
2:38
If the Determinant of a Matrix is Nonzero, the Matrix is Invertible
3:40
Determinant of a 2 x 2 Matrix
4:34
Think in Terms of Diagonals
5:12
Minors and Cofactors - Minors
6:24
Example
6:46
Minors and Cofactors - Cofactors
8:00
Cofactor is Closely Based on the Minor
8:01
Alternating Sign Pattern
9:04
Determinant of Larger Matrices
10:56
Example
13:00
Alternative Method for 3x3 Matrices
16:46
Not Recommended
16:48
Inverse of a 2 x 2 Matrix
19:02
Inverse of Larger Matrices
20:00
Using Inverse Matrices
21:06
When Multiplied Together, They Create the Identity Matrix
21:24
Example 1
23:45
Example 2
27:21
Example 3
32:49
Example 4
36:27
Finding the Inverse of Larger Matrices
41:59
General Inverse Method - Step 1
43:25
General Inverse Method - Step 2
43:27
General Inverse Method - Step 2, cont.
43:27
General Inverse Method - Step 3
45:15
Using Matrices to Solve Systems of Linear Equations

58m 34s

Intro
0:00
Introduction
0:12
Augmented Matrix
1:44
We Can Represent the Entire Linear System With an Augmented Matrix
1:50
Row Operations
3:22
Interchange the Locations of Two Rows
3:50
Multiply (or Divide) a Row by a Nonzero Number
3:58
Add (or Subtract) a Multiple of One Row to Another
4:12
Row Operations - Keep Notes!
5:50
Suggested Symbols
7:08
Gauss-Jordan Elimination - Idea
8:04
Gauss-Jordan Elimination - Idea, cont.
9:16
Reduced Row-Echelon Form
9:18
Gauss-Jordan Elimination - Method
11:36
Begin by Writing the System As An Augmented Matrix
11:38
Gauss-Jordan Elimination - Method, cont.
13:48
Cramer's Rule - 2 x 2 Matrices
17:08
Cramer's Rule - n x n Matrices
19:24
Solving with Inverse Matrices
21:10
Solving Inverse Matrices, cont.
25:28
The Mighty (Graphing) Calculator
26:38
Example 1
29:56
Example 2
33:56
Example 3
37:00
Example 3, cont.
45:04
Example 4
51:28
XI. Alternate Ways to Graph
Parametric Equations

53m 33s

Intro
0:00
Introduction
0:06
Definition
1:10
Plane Curve
1:24
The Key Idea
2:00
Graphing with Parametric Equations
2:52
Same Graph, Different Equations
5:04
How Is That Possible?
5:36
Same Graph, Different Equations, cont.
5:42
Here's Another to Consider
7:56
Same Plane Curve, But Still Different
8:10
A Metaphor for Parametric Equations
9:36
Think of Parametric Equations As a Way to Describe the Motion of An Object
9:38
Graph Shows Where It Went, But Not Speed
10:32
Eliminating Parameters
12:14
Rectangular Equation
12:16
Caution
13:52
Creating Parametric Equations
14:30
Interesting Graphs
16:38
Graphing Calculators, Yay!
19:18
Example 1
22:36
Example 2
28:26
Example 3
37:36
Example 4
41:00
Projectile Motion
44:26
Example 5
47:00
Polar Coordinates

48m 7s

Intro
0:00
Introduction
0:04
Polar Coordinates Give Us a Way To Describe the Location of a Point
0:26
Polar Equations and Functions
0:50
Plotting Points with Polar Coordinates
1:06
The Distance of the Point from the Origin
1:09
The Angle of the Point
1:33
Give Points as the Ordered Pair (r,θ)
2:03
Visualizing Plotting in Polar Coordinates
2:32
First Way We Can Plot
2:39
Second Way We Can Plot
2:50
First, We'll Look at Visualizing r, Then θ
3:09
Rotate the Length Counter-Clockwise by θ
3:38
Alternatively, We Can Visualize θ, Then r
4:06
'Polar Graph Paper'
6:17
Horizontal and Vertical Tick Marks Are Not Useful for Polar
6:42
Use Concentric Circles to Helps Up See Distance From the Pole
7:08
Can Use Arc Sectors to See Angles
7:57
Multiple Ways to Name a Point
9:17
Examples
9:30
For Any Angle θ, We Can Make an Equivalent Angle
10:44
Negative Values for r
11:58
If r Is Negative, We Go In The Direction Opposite the One That The Angle θ Points Out
12:22
Another Way to Name the Same Point: Add π to θ and Make r Negative
13:44
Converting Between Rectangular and Polar
14:37
Rectangular Way to Name
14:43
Polar Way to Name
14:52
The Rectangular System Must Have a Right Angle Because It's Based on a Rectangle
15:08
Connect Both Systems Through Basic Trigonometry
15:38
Equation to Convert From Polar to Rectangular Coordinate Systems
16:55
Equation to Convert From Rectangular to Polar Coordinate Systems
17:13
Converting to Rectangular is Easy
17:20
Converting to Polar is a Bit Trickier
17:21
Draw Pictures
18:55
Example 1
19:50
Example 2
25:17
Example 3
31:05
Example 4
35:56
Example 5
41:49
Polar Equations & Functions

38m 16s

Intro
0:00
Introduction
0:04
Equations and Functions
1:16
Independent Variable
1:21
Dependent Variable
1:30
Examples
1:46
Always Assume That θ Is In Radians
2:44
Graphing in Polar Coordinates
3:29
Graph is the Same Way We Graph 'Normal' Stuff
3:32
Example
3:52
Graphing in Polar - Example, Cont.
6:45
Tips for Graphing
9:23
Notice Patterns
10:19
Repetition
13:39
Graphing Equations of One Variable
14:39
Converting Coordinate Types
16:16
Use the Same Conversion Formulas From the Previous Lesson
16:23
Interesting Graphs
17:48
Example 1
18:03
Example 2
18:34
Graphing Calculators, Yay!
19:07
Plot Random Things, Alter Equations You Understand, Get a Sense for How Polar Stuff Works
19:11
Check Out the Appendix
19:26
Example 1
21:36
Example 2
28:13
Example 3
34:24
Example 4
35:52
XII. Complex Numbers and Polar Coordinates
Polar Form of Complex Numbers

40m 43s

Intro
0:00
Polar Coordinates
0:49
Rectangular Form
0:52
Polar Form
1:25
R and Theta
1:51
Polar Form Conversion
2:27
R and Theta
2:35
Optimal Values
4:05
Euler's Formula
4:25
Multiplying Two Complex Numbers in Polar Form
6:10
Multiply r's Together and Add Exponents
6:32
Example 1: Convert Rectangular to Polar Form
7:17
Example 2: Convert Polar to Rectangular Form
13:49
Example 3: Multiply Two Complex Numbers
17:28
Extra Example 1: Convert Between Rectangular and Polar Forms
-1
Extra Example 2: Simplify Expression to Polar Form
-2
DeMoivre's Theorem

57m 37s

Intro
0:00
Introduction to DeMoivre's Theorem
0:10
n nth Roots
3:06
DeMoivre's Theorem: Finding nth Roots
3:52
Relation to Unit Circle
6:29
One nth Root for Each Value of k
7:11
Example 1: Convert to Polar Form and Use DeMoivre's Theorem
8:24
Example 2: Find Complex Eighth Roots
15:27
Example 3: Find Complex Roots
27:49
Extra Example 1: Convert to Polar Form and Use DeMoivre's Theorem
-1
Extra Example 2: Find Complex Fourth Roots
-2
XIII. Counting & Probability
Counting

31m 36s

Intro
0:00
Introduction
0:08
Combinatorics
0:56
Definition: Event
1:24
Example
1:50
Visualizing an Event
3:02
Branching line diagram
3:06
Addition Principle
3:40
Example
4:18
Multiplication Principle
5:42
Example
6:24
Pigeonhole Principle
8:06
Example
10:26
Draw Pictures
11:06
Example 1
12:02
Example 2
14:16
Example 3
17:34
Example 4
21:26
Example 5
25:14
Permutations & Combinations

44m 3s

Intro
0:00
Introduction
0:08
Permutation
0:42
Combination
1:10
Towards a Permutation Formula
2:38
How Many Ways Can We Arrange the Letters A, B, C, D, and E?
3:02
Towards a Permutation Formula, cont.
3:34
Factorial Notation
6:56
Symbol Is '!'
6:58
Examples
7:32
Permutation of n Objects
8:44
Permutation of r Objects out of n
9:04
What If We Have More Objects Than We Have Slots to Fit Them Into?
9:46
Permutation of r Objects Out of n, cont.
10:28
Distinguishable Permutations
14:46
What If Not All Of the Objects We're Permuting Are Distinguishable From Each Other?
14:48
Distinguishable Permutations, cont.
17:04
Combinations
19:04
Combinations, cont.
20:56
Example 1
23:10
Example 2
26:16
Example 3
28:28
Example 4
31:52
Example 5
33:58
Example 6
36:34
Probability

36m 58s

Intro
0:00
Introduction
0:06
Definition: Sample Space
1:18
Event = Something Happening
1:20
Sample Space
1:36
Probability of an Event
2:12
Let E Be An Event and S Be The Corresponding Sample Space
2:14
'Equally Likely' Is Important
3:52
Fair and Random
5:26
Interpreting Probability
6:34
How Can We Interpret This Value?
7:24
We Can Represent Probability As a Fraction, a Decimal, Or a Percentage
8:04
One of Multiple Events Occurring
9:52
Mutually Exclusive Events
10:38
What If The Events Are Not Mutually Exclusive?
12:20
Taking the Possibility of Overlap Into Account
13:24
An Event Not Occurring
17:14
Complement of E
17:22
Independent Events
19:36
Independent
19:48
Conditional Events
21:28
What Is The Events Are Not Independent Though?
21:30
Conditional Probability
22:16
Conditional Events, cont.
23:51
Example 1
25:27
Example 2
27:09
Example 3
28:57
Example 4
30:51
Example 5
34:15
XIV. Conic Sections
Parabolas

41m 27s

Intro
0:00
What is a Parabola?
0:20
Definition of a Parabola
0:29
Focus
0:59
Directrix
1:15
Axis of Symmetry
3:08
Vertex
3:33
Minimum or Maximum
3:44
Standard Form
4:59
Horizontal Parabolas
5:08
Vertex Form
5:19
Upward or Downward
5:41
Example: Standard Form
6:06
Graphing Parabolas
8:31
Shifting
8:51
Example: Completing the Square
9:22
Symmetry and Translation
12:18
Example: Graph Parabola
12:40
Latus Rectum
17:13
Length
18:15
Example: Latus Rectum
18:35
Horizontal Parabolas
18:57
Not Functions
20:08
Example: Horizontal Parabola
21:21
Focus and Directrix
24:11
Horizontal
24:48
Example 1: Parabola Standard Form
25:12
Example 2: Graph Parabola
30:00
Example 3: Graph Parabola
33:13
Example 4: Parabola Equation
37:28
Circles

21m 3s

Intro
0:00
What are Circles?
0:08
Example: Equidistant
0:17
Radius
0:32
Equation of a Circle
0:44
Example: Standard Form
1:11
Graphing Circles
1:47
Example: Circle
1:56
Center Not at Origin
3:07
Example: Completing the Square
3:51
Example 1: Equation of Circle
6:44
Example 2: Center and Radius
11:51
Example 3: Radius
15:08
Example 4: Equation of Circle
16:57
Ellipses

46m 51s

Intro
0:00
What Are Ellipses?
0:11
Foci
0:23
Properties of Ellipses
1:43
Major Axis, Minor Axis
1:47
Center
1:54
Length of Major Axis and Minor Axis
3:21
Standard Form
5:33
Example: Standard Form of Ellipse
6:09
Vertical Major Axis
9:14
Example: Vertical Major Axis
9:46
Graphing Ellipses
12:51
Complete the Square and Symmetry
13:00
Example: Graphing Ellipse
13:16
Equation with Center at (h, k)
19:57
Horizontal and Vertical
20:14
Difference
20:27
Example: Center at (h, k)
20:55
Example 1: Equation of Ellipse
24:05
Example 2: Equation of Ellipse
27:57
Example 3: Equation of Ellipse
32:32
Example 4: Graph Ellipse
38:27
Hyperbolas

38m 15s

Intro
0:00
What are Hyperbolas?
0:12
Two Branches
0:18
Foci
0:38
Properties
2:00
Transverse Axis and Conjugate Axis
2:06
Vertices
2:46
Length of Transverse Axis
3:14
Distance Between Foci
3:31
Length of Conjugate Axis
3:38
Standard Form
5:45
Vertex Location
6:36
Known Points
6:52
Vertical Transverse Axis
7:26
Vertex Location
7:50
Asymptotes
8:36
Vertex Location
8:56
Rectangle
9:28
Diagonals
10:29
Graphing Hyperbolas
12:58
Example: Hyperbola
13:16
Equation with Center at (h, k)
16:32
Example: Center at (h, k)
17:21
Example 1: Equation of Hyperbola
19:20
Example 2: Equation of Hyperbola
22:48
Example 3: Graph Hyperbola
26:05
Example 4: Equation of Hyperbola
36:29
Conic Sections

18m 43s

Intro
0:00
Conic Sections
0:16
Double Cone Sections
0:24
Standard Form
1:27
General Form
1:37
Identify Conic Sections
2:16
B = 0
2:50
X and Y
3:22
Identify Conic Sections, Cont.
4:46
Parabola
5:17
Circle
5:51
Ellipse
6:31
Hyperbola
7:10
Example 1: Identify Conic Section
8:01
Example 2: Identify Conic Section
11:03
Example 3: Identify Conic Section
11:38
Example 4: Identify Conic Section
14:50
XV. Sequences, Series, & Induction
Introduction to Sequences

57m 45s

Intro
0:00
Introduction
0:06
Definition: Sequence
0:28
Infinite Sequence
2:08
Finite Sequence
2:22
Length
2:58
Formula for the nth Term
3:22
Defining a Sequence Recursively
5:54
Initial Term
7:58
Sequences and Patterns
10:40
First, Identify a Pattern
12:52
How to Get From One Term to the Next
17:38
Tips for Finding Patterns
19:52
More Tips for Finding Patterns
24:14
Even More Tips
26:50
Example 1
30:32
Example 2
34:54
Fibonacci Sequence
34:55
Example 3
38:40
Example 4
45:02
Example 5
49:26
Example 6
51:54
Introduction to Series

40m 27s

Intro
0:00
Introduction
0:06
Definition: Series
1:20
Why We Need Notation
2:48
Simga Notation (AKA Summation Notation)
4:44
Thing Being Summed
5:42
Index of Summation
6:21
Lower Limit of Summation
7:09
Upper Limit of Summation
7:23
Sigma Notation, Example
7:36
Sigma Notation for Infinite Series
9:08
How to Reindex
10:58
How to Reindex, Expanding
12:56
How to Reindex, Substitution
16:46
Properties of Sums
19:42
Example 1
23:46
Example 2
25:34
Example 3
27:12
Example 4
29:54
Example 5
32:06
Example 6
37:16
Arithmetic Sequences & Series

31m 36s

Intro
0:00
Introduction
0:05
Definition: Arithmetic Sequence
0:47
Common Difference
1:13
Two Examples
1:19
Form for the nth Term
2:14
Recursive Relation
2:33
Towards an Arithmetic Series Formula
5:12
Creating a General Formula
10:09
General Formula for Arithmetic Series
14:23
Example 1
15:46
Example 2
17:37
Example 3
22:21
Example 4
24:09
Example 5
27:14
Geometric Sequences & Series

39m 27s

Intro
0:00
Introduction
0:06
Definition
0:48
Form for the nth Term
2:42
Formula for Geometric Series
5:16
Infinite Geometric Series
11:48
Diverges
13:04
Converges
14:48
Formula for Infinite Geometric Series
16:32
Example 1
20:32
Example 2
22:02
Example 3
26:00
Example 4
30:48
Example 5
34:28
Mathematical Induction

49m 53s

Intro
0:00
Introduction
0:06
Belief Vs. Proof
1:22
A Metaphor for Induction
6:14
The Principle of Mathematical Induction
11:38
Base Case
13:24
Inductive Step
13:30
Inductive Hypothesis
13:52
A Remark on Statements
14:18
Using Mathematical Induction
16:58
Working Example
19:58
Finding Patterns
28:46
Example 1
30:17
Example 2
37:50
Example 3
42:38
The Binomial Theorem

1h 13m 13s

Intro
0:00
Introduction
0:06
We've Learned That a Binomial Is An Expression That Has Two Terms
0:07
Understanding Binomial Coefficients
1:20
Things We Notice
2:24
What Goes In the Blanks?
5:52
Each Blank is Called a Binomial Coefficient
6:18
The Binomial Theorem
6:38
Example
8:10
The Binomial Theorem, cont.
10:46
We Can Also Write This Expression Compactly Using Sigma Notation
12:06
Proof of the Binomial Theorem
13:22
Proving the Binomial Theorem Is Within Our Reach
13:24
Pascal's Triangle
15:12
Pascal's Triangle, cont.
16:12
Diagonal Addition of Terms
16:24
Zeroth Row
18:04
First Row
18:12
Why Do We Care About Pascal's Triangle?
18:50
Pascal's Triangle, Example
19:26
Example 1
21:26
Example 2
24:34
Example 3
28:34
Example 4
32:28
Example 5
37:12
Time for the Fireworks!
43:38
Proof of the Binomial Theorem
43:44
We'll Prove This By Induction
44:04
Proof (By Induction)
46:36
Proof, Base Case
47:00
Proof, Inductive Step - Notation Discussion
49:22
Induction Step
49:24
Proof, Inductive Step - Setting Up
52:26
Induction Hypothesis
52:34
What We What To Show
52:44
Proof, Inductive Step - Start
54:18
Proof, Inductive Step - Middle
55:38
Expand Sigma Notations
55:48
Proof, Inductive Step - Middle, cont.
58:40
Proof, Inductive Step - Checking In
1:01:08
Let's Check In With Our Original Goal
1:01:12
Want to Show
1:01:18
Lemma - A Mini Theorem
1:02:18
Proof, Inductive Step - Lemma
1:02:52
Proof of Lemma: Let's Investigate the Left Side
1:03:08
Proof, Inductive Step - Nearly There
1:07:54
Proof, Inductive Step - End!
1:09:18
Proof, Inductive Step - End!, cont.
1:11:01
XVI. Preview of Calculus
Idea of a Limit

40m 22s

Intro
0:00
Introduction
0:05
Motivating Example
1:26
Fuzzy Notion of a Limit
3:38
Limit is the Vertical Location a Function is Headed Towards
3:44
Limit is What the Function Output is Going to Be
4:15
Limit Notation
4:33
Exploring Limits - 'Ordinary' Function
5:26
Test Out
5:27
Graphing, We See The Answer Is What We Would Expect
5:44
Exploring Limits - Piecewise Function
6:45
If We Modify the Function a Bit
6:49
Exploring Limits - A Visual Conception
10:08
Definition of a Limit
12:07
If f(x) Becomes Arbitrarily Close to Some Number L as x Approaches Some Number c, Then the Limit of f(x) As a Approaches c is L.
12:09
We Are Not Concerned with f(x) at x=c
12:49
We Are Considering x Approaching From All Directions, Not Just One Side
13:10
Limits Do Not Always Exist
15:47
Finding Limits
19:49
Graphs
19:52
Tables
21:48
Precise Methods
24:53
Example 1
26:06
Example 2
27:39
Example 3
30:51
Example 4
33:11
Example 5
37:07
Formal Definition of a Limit

57m 11s

Intro
0:00
Introduction
0:06
New Greek Letters
2:42
Delta
3:14
Epsilon
3:46
Sometimes Called the Epsilon-Delta Definition of a Limit
3:56
Formal Definition of a Limit
4:22
What does it MEAN!?!?
5:00
The Groundwork
5:38
Set Up the Limit
5:39
The Function is Defined Over Some Portion of the Reals
5:58
The Horizontal Location is the Value the Limit Will Approach
6:28
The Vertical Location L is Where the Limit Goes To
7:00
The Epsilon-Delta Part
7:26
The Hard Part is the Second Part of the Definition
7:30
Second Half of Definition
10:04
Restrictions on the Allowed x Values
10:28
The Epsilon-Delta Part, cont.
13:34
Sherlock Holmes and Dr. Watson
15:08
The Adventure of the Delta-Epsilon Limit
15:16
Setting
15:18
We Begin By Setting Up the Game As Follows
15:52
The Adventure of the Delta-Epsilon, cont.
17:24
This Game is About Limits
17:46
What If I Try Larger?
19:39
Technically, You Haven't Proven the Limit
20:53
Here is the Method
21:18
What We Should Concern Ourselves With
22:20
Investigate the Left Sides of the Expressions
25:24
We Can Create the Following Inequalities
28:08
Finally…
28:50
Nothing Like a Good Proof to Develop the Appetite
30:42
Example 1
31:02
Example 1, cont.
36:26
Example 2
41:46
Example 2, cont.
47:50
Finding Limits

32m 40s

Intro
0:00
Introduction
0:08
Method - 'Normal' Functions
2:04
The Easiest Limits to Find
2:06
It Does Not 'Break'
2:18
It Is Not Piecewise
2:26
Method - 'Normal' Functions, Example
3:38
Method - 'Normal' Functions, cont.
4:54
The Functions We're Used to Working With Go Where We Expect Them To Go
5:22
A Limit is About Figuring Out Where a Function is 'Headed'
5:42
Method - Canceling Factors
7:18
One Weird Thing That Often Happens is Dividing By 0
7:26
Method - Canceling Factors, cont.
8:16
Notice That The Two Functions Are Identical With the Exception of x=0
8:20
Method - Canceling Factors, cont.
10:00
Example
10:52
Method - Rationalization
12:04
Rationalizing a Portion of Some Fraction
12:05
Conjugate
12:26
Method - Rationalization, cont.
13:14
Example
13:50
Method - Piecewise
16:28
The Limits of Piecewise Functions
16:30
Example 1
17:42
Example 2
18:44
Example 3
20:20
Example 4
22:24
Example 5
24:24
Example 6
27:12
Continuity & One-Sided Limits

32m 43s

Intro
0:00
Introduction
0:06
Motivating Example
0:56
Continuity - Idea
2:14
Continuous Function
2:18
All Parts of Function Are Connected
2:28
Function's Graph Can Be Drawn Without Lifting Pencil
2:36
There Are No Breaks or Holes in Graph
2:56
Continuity - Idea, cont.
3:38
We Can Interpret the Break in the Continuity of f(x) as an Issue With the Function 'Jumping'
3:52
Continuity - Definition
5:16
A Break in Continuity is Caused By the Limit Not Matching Up With What the Function Does
5:18
Discontinuous
6:02
Discontinuity
6:10
Continuity and 'Normal' Functions
6:48
Return of the Motivating Example
8:14
One-Sided Limit
8:48
One-Sided Limit - Definition
9:16
Only Considers One Side
9:20
Be Careful to Keep Track of Which Symbol Goes With Which Side
10:06
One-Sided Limit - Example
10:50
There Does Not Necessarily Need to Be a Connection Between Left or Right Side Limits
11:16
Normal Limits and One-Sided Limits
12:08
Limits of Piecewise Functions
14:12
'Breakover' Points
14:22
We Find the Limit of a Piecewise Function By Checking If the Left and Right Side Limits Agree With Each Other
15:34
Example 1
16:40
Example 2
18:54
Example 3
22:00
Example 4
26:36
Limits at Infinity & Limits of Sequences

32m 49s

Intro
0:00
Introduction
0:06
Definition: Limit of a Function at Infinity
1:44
A Limit at Infinity Works Very Similarly to How a Normal Limit Works
2:38
Evaluating Limits at Infinity
4:08
Rational Functions
4:17
Examples
4:30
For a Rational Function, the Question Boils Down to Comparing the Long Term Growth Rates of the Numerator and Denominator
5:22
There are Three Possibilities
6:36
Evaluating Limits at Infinity, cont.
8:08
Does the Function Grow Without Bound? Will It 'Settle Down' Over Time?
10:06
Two Good Ways to Think About This
10:26
Limit of a Sequence
12:20
What Value Does the Sequence Tend to Do in the Long-Run?
12:41
The Limit of a Sequence is Very Similar to the Limit of a Function at Infinity
12:52
Numerical Evaluation
14:16
Numerically: Plug in Numbers and See What Comes Out
14:24
Example 1
16:42
Example 2
21:00
Example 3
22:08
Example 4
26:14
Example 5
28:10
Example 6
31:06
Instantaneous Slope & Tangents (Derivatives)

51m 13s

Intro
0:00
Introduction
0:08
The Derivative of a Function Gives Us a Way to Talk About 'How Fast' the Function If Changing
0:16
Instantaneous Slop
0:22
Instantaneous Rate of Change
0:28
Slope
1:24
The Vertical Change Divided by the Horizontal
1:40
Idea of Instantaneous Slope
2:10
What If We Wanted to Apply the Idea of Slope to a Non-Line?
2:14
Tangent to a Circle
3:52
What is the Tangent Line for a Circle?
4:42
Tangent to a Curve
5:20
Towards a Derivative - Average Slope
6:36
Towards a Derivative - Average Slope, cont.
8:20
An Approximation
11:24
Towards a Derivative - General Form
13:18
Towards a Derivative - General Form, cont.
16:46
An h Grows Smaller, Our Slope Approximation Becomes Better
18:44
Towards a Derivative - Limits!
20:04
Towards a Derivative - Limits!, cont.
22:08
We Want to Show the Slope at x=1
22:34
Towards a Derivative - Checking Our Slope
23:12
Definition of the Derivative
23:54
Derivative: A Way to Find the Instantaneous Slope of a Function at Any Point
23:58
Differentiation
24:54
Notation for the Derivative
25:58
The Derivative is a Very Important Idea In Calculus
26:04
The Important Idea
27:34
Why Did We Learn the Formal Definition to Find a Derivative?
28:18
Example 1
30:50
Example 2
36:06
Example 3
40:24
The Power Rule
44:16
Makes It Easier to Find the Derivative of a Function
44:24
Examples
45:04
n Is Any Constant Number
45:46
Example 4
46:26
Area Under a Curve (Integrals)

45m 26s

Intro
0:00
Introduction
0:06
Integral
0:12
Idea of Area Under a Curve
1:18
Approximation by Rectangles
2:12
The Easiest Way to Find Area is With a Rectangle
2:18
Various Methods for Choosing Rectangles
4:30
Rectangle Method - Left-Most Point
5:12
The Left-Most Point
5:16
Rectangle Method - Right-Most Point
5:58
The Right-Most Point
6:00
Rectangle Method - Mid-Point
6:42
Horizontal Mid-Point
6:48
Rectangle Method - Maximum (Upper Sum)
7:34
Maximum Height
7:40
Rectangle Method - Minimum
8:54
Minimum Height
9:02
Evaluating the Area Approximation
10:08
Split the Interval Into n Sub-Intervals
10:30
More Rectangles, Better Approximation
12:14
The More We Us , the Better Our Approximation Becomes
12:16
Our Approximation Becomes More Accurate as the Number of Rectangles n Goes Off to Infinity
12:44
Finding Area with a Limit
13:08
If This Limit Exists, It Is Called the Integral From a to b
14:08
The Process of Finding Integrals is Called Integration
14:22
The Big Reveal
14:40
The Integral is Based on the Antiderivative
14:46
The Big Reveal - Wait, Why?
16:28
The Rate of Change for the Area is Based on the Height of the Function
16:50
Height is the Derivative of Area, So Area is Based on the Antiderivative of Height
17:50
Example 1
19:06
Example 2
22:48
Example 3
29:06
Example 3, cont.
35:14
Example 4
40:14
XVII. Appendix: Graphing Calculators
Buying a Graphing Calculator

10m 41s

Intro
0:00
Should You Buy?
0:06
Should I Get a Graphing Utility?
0:20
Free Graphing Utilities - Web Based
0:38
Personal Favorite: Desmos
0:58
Free Graphing Utilities - Offline Programs
1:18
GeoGebra
1:31
Microsoft Mathematics
1:50
Grapher
2:18
Other Graphing Utilities - Tablet/Phone
2:48
Should You Buy a Graphing Calculator?
3:22
The Only Real Downside
4:10
Deciding on Buying
4:20
If You Plan on Continuing in Math and/or Science
4:26
If Money is Not Particularly Tight for You
4:32
If You Don't Plan to Continue in Math and Science
5:02
If You Do Plan to Continue and Money Is Tight
5:28
Which to Buy
5:44
Which Graphing Calculator is Best?
5:46
Too Many Factors
5:54
Do Your Research
6:12
The Old Standby
7:10
TI-83 (Plus)
7:16
TI-84 (Plus)
7:18
Tips for Purchasing
9:17
Buy Online
9:19
Buy Used
9:35
Ask Around
10:09
Graphing Calculator Basics

10m 51s

Intro
0:00
Read the Manual
0:06
Skim It
0:20
Play Around and Experiment
0:34
Syntax
0:40
Definition of Syntax in English and Math
0:46
Pay Careful Attention to Your Syntax When Working With a Calculator
2:08
Make Sure You Use Parentheses to Indicate the Proper Order of Operations
2:16
Think About the Results
3:54
Settings
4:58
You'll Almost Never Need to Change the Settings on Your Calculator
5:00
Tell Calculator In Settings Whether the Angles Are In Radians or Degrees
5:26
Graphing Mode
6:32
Error Messages
7:10
Don't Panic
7:11
Internet Search
7:32
So Many Things
8:14
More Powerful Than You Realize
8:18
Other Things Your Graphing Calculator Can Do
8:24
Playing Around
9:16
Graphing Functions, Window Settings, & Table of Values

10m 38s

Intro
0:00
Graphing Functions
0:18
Graphing Calculator Expects the Variable to Be x
0:28
Syntax
0:58
The Syntax We Choose Will Affect How the Function Graphs
1:00
Use Parentheses
1:26
The Viewing Window
2:00
One of the Most Important Ideas When Graphing Is To Think About The Viewing Window
2:01
For Example
2:30
The Viewing Window, cont.
2:36
Window Settings
3:24
Manually Choose Window Settings
4:20
x Min
4:40
x Max
4:42
y Min
4:44
y Max
4:46
Changing the x Scale or y Scale
5:08
Window Settings, cont.
5:44
Table of Values
7:38
Allows You to Quickly Churn Out Values for Various Inputs
7:42
For example
7:44
Changing the Independent Variable From 'Automatic' to 'Ask'
8:50
Finding Points of Interest

9m 45s

Intro
0:00
Points of Interest
0:06
Interesting Points on the Graph
0:11
Roots/Zeros (Zero)
0:18
Relative Minimums (Min)
0:26
Relative Maximums (Max)
0:32
Intersections (Intersection)
0:38
Finding Points of Interest - Process
1:48
Graph the Function
1:49
Adjust Viewing Window
2:12
Choose Point of Interest Type
2:54
Identify Where Search Should Occur
3:04
Give a Guess
3:36
Get Result
4:06
Advanced Technique: Arbitrary Solving
5:10
Find Out What Input Value Causes a Certain Output
5:12
For Example
5:24
Advanced Technique: Calculus
7:18
Derivative
7:22
Integral
7:30
But How Do You Show Work?
8:20
Parametric & Polar Graphs

7m 8s

Intro
0:00
Change Graph Type
0:08
Located in General 'Settings'
0:16
Graphing in Parametric
1:06
Set Up Both Horizontal Function and Vertical Function
1:08
For Example
2:04
Graphing in Polar
4:00
For Example
4:28
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Lecture Comments (3)

2 answers

Last reply by: Duy Nguyen
Thu Aug 6, 2015 10:24 PM

Post by Duy Nguyen on August 6, 2015

Hi, would you mind explaining why the domain of a log function does not include negative numbers? Because log base (-2) of (-8) is 3 and x, in this case, is a negative number.

Thank you very much.

Introduction to Logarithms

  • A logarithm is a way to reverse the process of exponentiation. It allows us to mathematically ask the question, "Given some base and some value, what exponent would we have to use on the base to create that value?"
  • The logarithm base a of x (loga x), where a > 0 and a ≠ 1, is defined to be the number y such that ay = x.
    loga x = y     ⇔     ay = x.
  • The idea of a logarithm can be really confusing the first few times you work with it, so make sure to watch the video to clarify how logarithms are used and what they mean.
  • The two most common logarithmic bases to come up are the numbers e and 10. As such, they have special notation because we have to write them so often.
    • The base of e is expressed as ln. It is called the natural logarithm. [Remember, e is called the natural base.]
      lnx     ⇔     loge x
    • The base of 10 is expressed with just log: if no base is given, it is assumed to be base 10. It is called the common logarithm.
      logx     ⇔     log10 x
  • Just like exponentiation, we can find the value (or a very good approximation) of a logarithm by using a calculator. Any scientific or graphing calculator will have ln and log buttons to take logarithms base e and 10, respectively. However, many calculators will not have a way to take logarithms of arbitrary bases. There is a way around this called change of base, and we'll explore it in the next lesson, Properties of Logarithms.
  • When we graph logarithms, we see they grow very slowly. (This is because they are the inverse of exponentiation.) The graph of a logarithm also approaches the y-axis asymptotically. It gets very close, but it doesn't touch it.
  • Logarithms are the inverse of exponentiation, and vice-versa. The exponential function of base a is the inverse of the logarithmic function of base a:
    f(x) = loga x               f−1(x) = ax
  • Because a logarithm is the inverse of exponentiation, we cannot take the logarithm of some numbers. Exponentiation (ax) only has a range of (0,∞), so the corresponding logarithm can only reverse those same output values. In other words,
    Domain of loga x:    (0, ∞) .
  • We can also see the above domain must be true because it would make no sense otherwise. Consider that log2 0 = b means that 2b = 0. But no such number b exists that could do this! The same goes for negative numbers, so the domain of any logarithm must be (0, ∞).
    Note: This means we have now introduced a new type of thing to watch out for when we're looking for domains. Before, we only had to worry about dividing by 0 and having negatives under a square root. (Depending on what you've done previously, you might also have needed to be careful about certain things with trigonometric functions.) Now we also have to watch out for taking a logarithm of 0 or a negative number. One more thing to pay attention to when you're finding the domain of a function.

Introduction to Logarithms

Write the logarithmic equation below in exponential form.
log5 125 = 3
  • The logarithm for the base a and some number x is defined as follows:
    loga x = y     ⇔     ay = x
    In other words, the logarithm of a number gives the appropriate exponent (for the log's base) to create the number (using that base) that the logarithm is acting upon.
  • Thus, we can follow this swapping procedure to see an alternate, exponential way to write the equation given in the question:
    log5 125 = 3     ⇔     53 = 125
    [Notice that you can visualize it as if the base of the logarithm "slides" under the number that is opposite the logarithm.]
53 = 125
Evaluate (without a calculator): log4 16
  • The logarithm for the base a and some number x is defined as follows:
    loga x = y     ⇔     ay = x
    In other words, the logarithm of a number gives the appropriate exponent (for the log's base) to make the number (using that base) that the logarithm is acting upon.
  • With this definition in mind, we see that evaluating the logarithm is the same as figuring out what number the question mark would be below:
    log4 16 =  ?     ⇔     4? = 16
  • Therefore, all we need to do is figure out what exponent on 4 will create 16. Pretty quickly we see that 42 = 16, so we have that log4 16 = 2.
2
Evaluate (without a calculator): log2 16
  • The logarithm for the base a and some number x is defined as follows:
    loga x = y     ⇔     ay = x
    In other words, the logarithm of a number gives the appropriate exponent (for the log's base) to make the number (using that base) that the logarithm is acting upon.
  • With this definition in mind, we see that evaluating the logarithm is the same as figuring out what number the question mark would be below:
    log2 16 =  ?     ⇔     2? = 16
  • Therefore, all we need to do is figure out what exponent on 2 will create 16. We start working through exponents of 2:
    20 = 1        21 = 2        22 = 4        23 = 8        24 = 16
    Thus, we see that 24=16, so we have that log2 16 = 4.
4
Evaluate (without a calculator): log25 [1/5]
  • The logarithm for the base a and some number x is defined as follows:
    loga x = y     ⇔     ay = x
    In other words, the logarithm of a number gives the appropriate exponent (for the log's base) to make the number (using that base) that the logarithm is acting upon.
  • With this definition in mind, we see that evaluating the logarithm is the same as figuring out what number the question mark would be below:
    log25 1

    5
    = ?     ⇔     25? = 1

    5
  • Therefore, we need to figure out what exponent on 25 will create [1/5].
    250 = 1        251 = 25        252 = 625
    Clearly, a positive exponent will not allow us to create the fraction we're looking for. However, we can try negative exponents to make smaller values closer to 0.
  • Trying negative exponents out, we get:
    250 = 1        25−1 = 1

    25
           25−2 = 1

    625
    Since [1/5] is somewhere between 1 and [1/25], we see that we need an exponent somewhere between 0 and 1.
  • At this point, we hopefully remember that a fractional exponent has the same effect as a radical (see the lesson on the properties of exponents if this is unfamiliar to you):
    a[1/n] =
    n
     

    a
     
           ⇒        25[1/2] =

     

    25
     
    = 5
    Therefore, if we combine having a negative exponent with a fractional exponent, we can achieve the value we're looking for:
    25−[1/2] = 1

    5
    ,
    therefore log25 [1/5] = −[1/2].
−[1/2]
Evaluate (without a calculator): lne47
  • First, it's important to realize that  ln is just a shorthand way of writing  loge. Thus, the question we're dealing with could be rephrased as
    lne47     ⇒     loge e47
  • The logarithm for the base a and some number x is defined as follows:
    loga x = y     ⇔     ay = x
    In other words, the logarithm of a number gives the appropriate exponent (for the log's base) to make the number (using that base) that the logarithm is acting upon.
  • With this definition in mind, we see that evaluating the logarithm is the same as figuring out what number the question mark would be below:
    loge e47 =  ?     ⇔     e? = e47
  • Once written out in this format, it's quite easy to see what exponent on e would create e47: clearly, 47 is the only reasonable choice! Thus, we see that our logarithm must give that as the result.
    lne47 = 47
47
Write the exponential equation below in logarithmic form.
75 = 16807
  • The logarithm for the base a and some number x is defined as follows:
    loga x = y     ⇔     ay = x
    In other words, the logarithm of a number gives the appropriate exponent (for the log's base) to make the number (using that base) that the logarithm is acting upon.
  • Thus, we can follow this swapping procedure to see an alternate, logarithmic way to write the equation given in the question:
    75 = 16807     ⇔     log7 16807 = 5
    [Notice that you can visualize it as if the base of the exponential expression "slides" out and becomes the base of the logarithm on the other side.]
log7 16807 = 5
Write the exponential equation below in logarithmic form.
e−4.2 = 0.014 995 …
  • The logarithm for the base a and some number x is defined as follows:
    loga x = y     ⇔     ay = x
    In other words, the logarithm of a number gives the appropriate exponent (for the log's base) to make the number (using that base) that the logarithm is acting upon.
  • Thus, we can follow this swapping procedure to see an alternate, logarithmic way to write the equation given in the question:
    e−4.2 = 0.014 995 …    ⇔     loge 0.014 995 … = −4.2
    [Notice that you can visualize it as if the base of the exponential expression "slides" out and becomes the base of the logarithm on the other side.]
  • Finally, when writing loge, we typically use the shorthand version written as  ln instead. [loge is still also correct, just less common.]
ln0.014 995 … = −4.2
Graph f(x) = log4 x.
  • When working out the graph of this function, remember that we can think of a logarithm as figuring out the question mark for the below:
    log4 x =  ?     ⇔     4? = x
    With this in mind, we realize the below:
    log4 1 = 0       log4 4 = 1        log4 16 = 2        log4 64 = 3
  • When graphing any function, it helps to make a table of values so we can plot points. With logarithms, it makes it a lot easier for us if we choose our x-values so the the logarithm will produce "clean" results, like we saw above. Working on our table, we find:
    x
    f(x)
    1
    0
    4
    1
    16
    2
    64
    3
    256
    4
  • Now that we know what happens as we go to the right, what about as we go the left? At this moment, we need to realize that logarithms only have a domain of (0, ∞). [Check out the video lesson for a detailed explanation of why this is the case.] Because of this, the function simply does not exist for x=0 or less.
  • However, the function does exist for 0 < x < 1. So what happens for those small fractions? Notice that [1/4] = 4−1. Thus, log4 [1/4] = −1. This process continues, and we can add it to our table of values:
    x
    f(x)
    1/4
    −1
    1/16
    −2
    1/64
    −3
    1/256
    −4
    Thus, we see that the function has a vertical asymptote at x=0, shooting down to −∞ as x becomes very, very small.
  • Finally, we're ready to put this all together and draw the graph. In choosing the axes, it is useful to choose a very large set of x-values for a comparatively small set of y-values because the function takes so long to grow. Plot points, then connect with curves.
Give the domain and range of the below function.
log3 (−x+5)
  • First, let's deal with the domain. (Remember, the domain of a function is the set of all input values it can accept.) For any logarithm, regardless of the base, it can accept the set of values (0,  ∞). In this case, though, it's not simply logx, so the domain is not simply (0, ∞).
  • Notice that the logarithm is acting on (−x+5), so what we need to do is figure out when the quantity (−x+5) outputs something in the set (0,  ∞). That is, when is the below inequality true?
    (−x+5) > 0
  • This inequality is true when
    x < 5,
    so we have a domain that is x: (−∞, 5).
  • Now that we know the domain, we need to figure out the range. (Remember, the range of a function is the set of all values it can output.) If we look at the graph of a standard logarithmic function (such as the one we worked out in the previous question), we see that the function can output values of (−∞,  ∞). [Why is this the case? A logarithm is basically a way of figuring out what the value of the question mark in an expression like 10? = number. Since there exists a number for any value you choose to plug in for the question mark, it's possible that the question mark can be any value. Thus, it's possible for the logarithm to output any value.]
  • With the specific function we're working on in this problem, this range will still hold true. While it's not precisely a standard logarithmic function because the logarithm is acting on (−x+5), we haven't affected the values that the logarithm will output in any way. Thus, the range is (−∞,  ∞).
Domain: (−∞, 5),    Range: (−∞,  ∞)
Give the domain and range of the below function.
g(x) =
ln(x−19)
2
 
+ 7
  • First, let's deal with the domain. (Remember, the domain of a function is the set of all input values it can accept.) For any logarithm, regardless of the base, it can accept the set of values (0,  ∞). In this case, though, it's not simply lnx, so the domain is not simply (0, ∞).
  • Notice that the logarithm is acting on (x−19), so what we need to do is figure out when the quantity (x−19) outputs something in the set (0,  ∞). That is, when is the below inequality true?
    (x−19) > 0
  • This inequality is true when
    x > 19,
    so we have a domain that is x: (19,  ∞).
  • Now that we know the domain, we need to figure out the range. (Remember, the range of a function is the set of all values it can output.) If we look at the graph of a standard logarithmic function (such as the one we worked out in the previous question), we see that a normal logarithmic function can output values of (−∞,  ∞). However, for the function we're working on, that won't be the case. But it's a useful starting place to keep in mind.
  • Notice that the logarithm is being squared: [ln(x−19) ]2. Thus, while ln(x−19) can output (−∞,  ∞), once it is squared, it can no longer produce negatives. Thus [ln(x−19) ]2 can only output [0,  ∞). Next, notice that the function then adds 7 to that:
    g(x) =
    ln(x−19)
    2
     
    + 7
    Therefore, the output of [ln(x−19) ]2 is "raised up" by 7, resulting in a range of [7,  ∞) for g(x).
Domain: (19,  ∞),    Range: [7,  ∞)

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

Introduction to Logarithms

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Introduction 0:04
  • Definition of a Logarithm, Base 2 0:51
    • Log 2 Defined
    • Examples
  • Definition of a Logarithm, General 3:23
  • Examples of Logarithms 5:15
    • Problems with Unusual Bases
  • Shorthand Notation: ln and log 9:44
    • base e as ln
    • base 10 as log
  • Calculating Logarithms 11:01
    • using a calculator
    • issues with other bases
  • Graphs of Logarithms 13:21
    • Three Examples
    • Slow Growth
  • Logarithms as Inverse of Exponentiation 16:02
    • Using Base 2
    • General Case
    • Looking More Closely at Logarithm Graphs
  • The Domain of Logarithms 20:41
    • Thinking about Logs like Inverses
    • The Alternate
  • Example 1 25:59
  • Example 2 30:03
  • Example 3 32:49
  • Example 4 37:34

Transcription: Introduction to Logarithms

Hi--welcome back to Educator.com.0000

Today, we are going to have an introduction to logarithms.0002

At this point, if we want to find the value of a number raised to an exponent, it is easy.0005

We use our exponentiation rules and evaluate; if it is something simple, like 23,0008

then we know that is 2 times 2 times 2, 2 times itself 3 times, which we figure out is 8.0012

And we also figured out how to do numbers that weren't just simple integer exponents.0018

But we have all of these nice rules from our previous few lessons.0022

But what if the question was inverted, and what if we knew the base and the end result, but we don't know the exponent that we need to get there?0025

If we knew that we had 2 as the base, and we wanted to have an end result of 8,0033

but we had no idea what exponent we had to use to get to 8, how could we figure out that exponent?0038

This is the question that leads us to explore the idea of the logarithm, which we will be looking at over the next few lessons.0045

We define the logarithm as a way to talk about this unknown exponent.0052

The logarithm base 2 of x, denoted log2(x), with this little 2 as a subscript right here,0055

is defined to be the number y such that 2y = x.0065

So, when we see log2(x) = y, then we know that that would be 2y = x.0071

In other words, we are looking for the number that...when the logarithm goes onto some number,0079

we are trying to figure out what value, if it raised to this space here, would become the number that we took the logarithm of.0085

So, for the example of log2(8) = 3, the reason why that is the case is because0094

we are looking for what number here we have to raise to to get 8.0100

So, the answer there is 3; if we raise 23, we get 8, so we know that log2(8) = 3.0105

It can be a little bit confusing to remember how this works at first--what the notation means.0115

So, when you see log2(x) = y, you can think of this as...if the 2 were under the y,0120

if we had 2y, it would make the x that we are taking the logarithm of.0129

So, we take the logarithm of some number in regards to some base.0135

And that tells us what number we would raise the base to, to get the original number we are taking the logarithm of.0139

It will make more sense as we see more examples.0146

Here are some other examples: log2(32) = 5, because if we raise 2 to the 5th, we get 32.0149

2 times 2 times 2 times 2 times 2 is equal to 32; 4, 8, 16, 32.0158

log2(1) = 0 because, if we raised 2 to the 0, from our rules about exponents, we know that that is just the same thing as 1.0168

log2(1/4) = -2, because we know that, if we raise 2 to the -2, then that is the same thing as 1/2 raised to the 2,0177

where you flip to the reciprocal; and 1/2 squared would become 1/4, which is what we initially started with.0186

So, a logarithm is a way of taking a logarithm of a number, so that you figure out0194

what you would have to raise some base to, to get the thing you took the logarithm of.0199

We can expand this idea to something beyond just base 2, to a general idea.0204

The logarithm base a of x, loga(x) (that little a, right down here, is a subscript), where a > 0,0209

and a is not equal to 1 (our base has to be greater than 0, and our base has to not be equal to 1),0218

is defined to be the number y such that ay = x.0224

If we take loga(x), then we know that that gives us some y, and that ay = x.0228

So, once again, it is the same idea, where, if we take this base, and we put it under the y, we would get ay.0236

And then, we would have the thing that we were originally taking the logarithm of, which is what we have there.0245

That is what is occurring right here.0249

The idea of the logarithm is that you take the log, and it tells you something that you can raise a number to, to be able to get this other value.0251

It is a little bit complex the first time you get it; but as you do it more and more, it will start to make more sense.0260

And we are going to see a whole bunch of examples to really get this cleared up.0264

Now, notice: we have these restrictions on what our base can be.0267

We know that the base has to be greater than 0, and the base is not equal to 1.0271

The base a of a logarithm has the same restrictions as the base of an exponential function.0274

This is because exponentiation and logarithms are inverse processes; they do the opposite thing.0280

And we will see more about how they are inverses in the future.0288

But they do reverse things, so they have to have the same restrictions, because they are basically working with the same idea of a base.0291

They are being seen through different lenses, and it will make more sense as we work on it more and more;0299

but we have to have the same restrictions on it--otherwise the idea of a logarithm will just not make sense, or not be very interesting.0303

So, we have these restrictions: that we have to have our base greater than 0, and we have to have our base not equal to 1.0309

Let's look at some examples to help clear this idea up.0316

log7(49) = 2, because, if we move this base over,0319

then 72 is just 7 times 7, which gets us 49, which is exactly what we started with.0324

So, this is the case, because 72 = 49; or maybe let's write it in the way that we had it originally here: 49 =...0332

moving the base over, moving our base underneath the right side...we have 72, like this.0343

The same thing over here: we know that log10(10000) = 4,0350

because, if we move our base under, we know that 104 is equal to 10000.0354

The question is: if we want to know what number we have to raise 10 to, to get 10000--that is what log10(10000) is effectively asking.0363

It is saying, "What number do we have to raise 10 to--raise our base to--to get 10000 as the end result?"0379

10 to the what equals 10000? The answer to that is 4.0387

So, we take 104; we get 10000; and sure enough, we see that that is 10 times 10 (100) times 10 (1000) times 10 (10000).0391

The same thing is going on over here: if we have log5(1/125), then we move that under, and we get 5-3 = 1/125.0400

Let's check that out: 5-3 is the same thing as 1/5 to the positive 3; we flip to the reciprocal.0413

And then, 5 times 5 times 5 is 25...125; so we get 1/125; sure enough, it checks out.0423

Finally, if we take log4(2), then we see that 41/2 is equal to 2.0433

What is 1/2? 1/2 is an exponent that means square root; 41/2 is the same thing as √4, which is 2; so once again, that checks out.0441

It is the question of what exponent I am looking for to be able to get this base to become the number that I am taking the logarithm of.0451

We can even do this with more unusual bases on our logarithms.0459

For example, if we have log1/2(1/16), then we can see that that will become 4, because (1/2)4 is equal to 1/16,0462

because 2, 4, 8, 16...1/2, 1/4, 1/8, 1/16; so we see that that is the same thing.0475

If we take logπ(π), then we know that it has to come out as 1,0486

because π1--of course, that is no surprise that it is going to equal what it already started with.0492

So, if we are taking logπ(π), then the thing that π has to be raised to, to get π, is just 1.0499

So, we have 1 as the thing that comes out of that.0505

If we take log√2(4√2), then we get that that has to be 5, because (√2)5--sure enough, that is equal to 4√2.0509

We can check that out: √2 times √2 times √2 times √2 times √2.0524

Well, √2 times √2 becomes 2; √2 times √2 becomes 2; and we have this one √2 left.0531

So, 2 times 2 is 4, and we are left with 4√2; it checks out.0537

The final one: loge(1) becomes 0, because if we move this over, e0, just like anything raised to the 0, becomes 1.0542

So, it checks out; so we have some idea of how it works.0553

A logarithm is, "For this base, what number do I have to raise it to, to get the number that I am originally taking the logarithm of?"0555

When I take the log10(10000), it is a question of what number I have to raise 10 to, to get 10000.0562

I have to raise 10 to 4 to get 10000.0570

When I take log7(49), it is a question of what number I need to raise 7 to, to get 49; the answer to that is 2.0574

That is the idea of a logarithm.0582

The two most common logarithmic bases to come up are the numbers e (remember, e is the natural number;0585

we talked about it previously, when we talked about exponential functions--a very important idea) and 10.0591

As such, they have special notation, because we have to write them so often.0597

The base of e is expressed as ln; so when we want to talk about base of e, the shorthand for that is ln.0601

It is called the natural logarithm; remember, e is called the natural base.0607

So, when we are taking a loge, we call it a natural logarithm, and we use ln, because originally,0612

the French were the ones who came up with this; so it was logarithme naturel (excuse my French--I am not very good at speaking French).0617

So, the natural log of x is equivalent to loge(x); ln(x) is just a shorthand way of saying log with a base of the number e.0623

Base 10 is expressed with just log on its own; notice, it has no subscript--there is no little number down there.0634

If no base is given, it is assumed to be base 10; since base 10 comes up a lot, it is just an easy way to write it; this is normally what it means.0641

It is called the common logarithm, because it is a commonly-used logarithm.0648

So, if you see log(x), notice that it has no little subscript--no little number down there.0652

Then, we know that that is going to mean log10(x).0657

Well, we can find the value of expressions like log2(8); we know that that came out to be 3,0662

because the number that we raised 2 to, to get 8, is 3.0667

How do we figure out the value of more complicated expressions?0670

Like if we wanted to figure out the natural log of 12.19--and as we just saw, that would be the same thing as asking, "What is loge or 12.19?"0673

Well, e is a very complicated number; it goes on forever--it is irrational.0681

12.19 is not a very friendly--looking decimal number; so how are these two things going to interact?0685

We can guess that it is probably not going to come out very cleanly, in a nice way.0690

Sure enough, it doesn't: it comes out to be approximately 2.500616; and precisely, it would keep going forever, as well.0693

So, for calculating logarithms, just like exponentiation, we can find the expressions, or a very good approximation, by using a calculator.0701

Any scientific or graphing calculator will have natural log and log10 buttons to take logarithms base e and 10, respectively.0709

However, many calculators will not have a way to take logarithms of arbitrary bases.0718

So, if we had log3, most calculators won't have an easy way for us to just get what log3(some number) is.0723

But there is a way around this, and it is called change of base.0731

So, if you do need to take the log base 3 of some number, check out the next lesson, Properties of Logarithms,0734

where we will explore how you can change from one base to another.0740

So, the way that you calculate complicated logarithms like this is: you generally just use a calculator.0743

The calculator has a way, a method, to be able to figure out what that comes out to be.0748

Now, just like with exponentiation that we talked about before, we should note that there are ways to calculate these values by hand.0753

We could do this by hand and figure it out; and you will learn about this in more advanced math classes.0759

But we won't learn about it in this course right here.0764

Doing this takes a lot of arithmetic, though; and so we designed calculators to speed up the process.0766

It is something that we could do; it is not like we are completely reliant on calculators for figuring this idea out.0771

Logarithms weren't something that we only got once we had calculators created.0776

We have been able to have this idea for a very long time--since the 1600s, in fact.0780

But being able to calculate what these numbers come out to be--that takes a long time; it is a slow process.0785

So, we have calculators to be able to figure this out for us very quickly and very easily.0790

So, it speeds things up, but it is not that we are dependent on calculators.0794

It is just that they are a useful tool that we can apply in this situation.0797

Graphs of logarithms: so now, since we can evaluate logarithms however we want,0802

because we have these nice calculators as tools, we can plot graphs of them.0806

So, let's look at some graphs: f(x) = log2(x) (this is in red); g(x) = log5(x)0809

(that one is in blue), and finally h(x) = log10(x) (that one is in green).0817

Notice how short the y-axis is; it only goes from -3 up until positive 5.0823

But we go all the way out to positive 100 on the x-axis.0830

We can see that here--right here; it is hard to see--that is a 1 value on the x-axis.0835

And that is going to end up corresponding to 0, because log of anything--log of any base of 1--will come out to be 0,0842

because the number that you raise anything to, a0, = 1.0855

So, if we would take log base anything of 1, it is going to always come out to be 0,0859

because that is the number that we raise anything to, to get 1 in the first place.0867

So, that is why we see a common height of 0 there.0871

And notice how slowly they grow: at 16, log2 is only going to be at a meager 4.0873

But for log10, when we look at log10, it takes getting all the way up to 100 to even get to 2.0881

If we go out here to the 2, it takes all the way to 100 to be able to get that from log10, because 102 = 100.0889

We are seeing a similar thing for log5: it has to get all the way up to 25 before it hits this height of 2, as well, because it is 52.0899

And we aren't even going to see it hit height 3, because it is not going to hit a height of 3 until it manages to get to 1250909

as an input value, because 53 becomes 125.0915

So, notice how slowly these graphs grow.0919

These graphs grow really, really slowly, because for logs, it takes a really big number to be able to get even slight increases in our verticals.0923

And the farther out they go, the even slower they are going to grow.0930

Now, notice that they approach the y-axis asymptotically.0933

So, as they get smaller and smaller, they get really, really close to this y-axis right here.0937

They never touch or pass it, although that might be hard to see in this picture, since it looks like it is right on top of it.0943

But they get very close; they won't actually touch it, but they get very close to the y-axis.0948

We will talk about this behavior of how it gets really close to the y-axis,0954

and why it can't actually touch the y-axis or go past it, soon, when we talk about the domain of a logarithm.0957

The logarithm is the inverse process of exponentiation.0963

For example, let's consider base 2: if we have log2(x) = y, then we can see its flip of 2y = x.0966

We just change the x and the y location.0973

So, if we take log2(8), that becomes 3, because remember: 23 = 8.0975

So, when we take log2(8), we get 3.0982

But then, if we take that 3, and we plug it into the other one--we take the 3, and we plug it in up here--0985

we look at 23--look: we are right back where we started.0990

We have the same thing on both sides.0993

We take this log, and we do something to it, and then we do the reverse process with the same base as the exponentiation.0997

We get back to the original input that we put in.1003

The same thing: we did it the other way, where we did exponentiation first.1006

If we take 2-2, then that is going to flip to (1/2)2; so we would get 1/4.1009

And then, if we take log2(1/4), we are going to get -2.1013

So, exponentiation and logarithms are doing inverses: one goes one way, and one goes the other way.1018

Together, they cancel out; we will be discussing this idea a lot more in the coming lessons.1023

It is a very important thing; we will also be proving it in general.1027

We can see this as one in a general form for any logarithm.1030

The exponential function of base a is the inverse of the logarithmic function of base a.1034

It is critical, though, that they do have the same bases; our exponential function is base a, and our logarithmic function must be base a.1039

If they are not the same base, it won't work.1046

Let's see why this is the case: if we have f(x) = loga(x), f-1(x) = ax.1048

Then, we can take f-1(f(x)) and see what happens.1055

Now, remember: we are talking about stuff from our lesson on inverse functions.1061

If you need more background on inverse functions, make sure you go and check out that lesson.1065

It will help you understand what is going on here.1068

So, f-1(f(x)) =...well, we could do this as...since this is ax, then it is going to be...1071

well, we will apply f-1 next; first, f(x) = loga(x).1077

We have loga(x); then we apply the f-1, and we have aloga(x).1082

Now, what does that end up coming out to be?1092

Well, remember: loga(x) = y is the same thing as saying ay = x.1094

So, that is what loga(x) is: it is this y, some y such that if we were to put it as an exponent on a, we would get x.1104

So, loga(x) = y: we can just say, "Whatever the number loga(x) is, let's call it y."1115

So, we can swap that out, and we can say, "ay," just because we are saying we will call loga(x) y.1120

That is what we have over here; but remember, we defined this idea of what loga(x) is based on ay = x.1126

Well, we now have ay = x; so if ay = x, then that equals x,1133

which means that f-1(f(x)) = x.1139

Whatever we put in as our input comes out as our output if we do these two functions, one on top of the other.1144

We have inverse functions, because one function cancels out the effects of the other function.1149

We will talk about this more in future lessons.1154

We can also see this in the graphs of exponential and logarithmic functions.1157

So, if we take two graphs of, say, 2x (that one is in red) and log2(x), we see them like this.1161

And then finally, we also have y = x in yellow here, coming through the middle.1168

Now, remember from our lesson about inverse functions: when we learned about inverse functions,1174

we know that if two functions are inverses, they mirror over the line y = x.1180

They are swapping x and y coordinates; this shows us that they have to be inverses.1188

For example, if we look at what log2 at 2 is, it comes out to be a height of 1; here is a height of 1.1193

And then, if we look at what our 2x at 1 is, at 1 it is a height of 2.1201

So, for this one, we have (I'll color-code it back to what it had been) (1,2).1212

But for the blue one, we have (2,1).1223

They flip x and y locations, and that is going to be true wherever we go on this, because we see that they do this thing with y = x,1226

where they mirror across it; their x and y locations swap, showing us that they are inverses.1236

Notice all the graphs that we have seen of logarithms: they never pass, or even touch, the y-axis.1243

They never pass the y-axis; they never even manage to touch the y-axis.1248

This is because the domain of a logarithm is 0 to infinity.1252

And notice: there is a parenthesis on that 0; so it says it is not inclusive--1257

so, not including 0, everywhere up from 0 (but not including 0), all the way up to positive infinity.1260

We can see this for a couple of reasons.1266

First, since exponentiation and logarithms are inverses, that means that the range of an exponential function is the domain of a logarithm.1268

The range of f(x) = ax is going to be 0 to infinity.1275

ax...if we put in any base a that is greater than 0 and not 1, it is going to go anywhere from 0 up until infinity.1280

If we look at 2x, by varying what we plug in for x, we are going to be able to get anything between 0 and positive infinity.1288

Now, let's talk briefly about this idea: if we had a pool of numbers that we called a, the set of things that we are allowed to use,1297

and then we had another pool of numbers that was b, the set of things that it is possible to get to through some function f...1306

we have some function f that takes numbers from a, and it goes to b; then we call the numbers over here domain.1315

We talked about this when we first talked about the ideas of functions.1322

So, here is the domain of f; and over here is the range of f.1326

The domain of f is everything that f is able to take in; the range of f is everything that f is able to put out.1331

So, for the example ax or the example 2x, as a specific example,1338

the domain is anything; it can take in any number at all--negative infinity to positive infinity--any real number whatsoever.1343

But it is only going to be able to give out numbers from 0 to infinity.1352

So, in this case, we see that it is going to have its range as 0 to infinity.1355

Now, notice: if we do the reverse of this, if we want to see the reverse of this, a function that does the opposite of what f does,1359

f-1, then it is going to have to go, not from a, but from b.1368

So, its domain, the domain of f-1, is going to be going in the other direction.1373

Since it is taking what f did and reversing it, it has to be able to take the things that f does as outputs.1380

Whatever f makes as outputs--whatever f puts out--is what f-1 will take in.1385

So, the domain of f-1 is the range of f, which means that the range of f-1 is also the domain of our original function, f.1391

f goes from a to b; f-1 goes from b to a.1402

Now, we saw that, for any exponential function, its range is 0 to infinity.1407

So, that means that the domain of f-1, the domain of a logarithmic function,1412

since it is the inverse of exponentiation, must also be from 0 to infinity.1419

So, that is why we have this domain here; the domain of any log has to be from 0 to infinity,1425

because the range of any exponential function is from 0 to infinity.1430

So, they are going to be done as opposites; the range of an exponential function is the domain of a logarithmic function.1434

So, that is a fancy way to be able to understand why this has to be the case,1441

because we can say what we learned about inverse functions applies here, because we have an inverse function.1446

But alternatively, we can just see that it would not make sense--it just is nonsense if we look at it otherwise.1450

Consider if we tried to take log2(0); then we know that that has to be equal to some number b for it to be a possible thing.1456

Then that means that 2b has to somehow be equal to 0.1463

But that doesn't make any sense: no such number b exists.1470

No possible number could exist that would be able to take 2 and turn it into 0.1474

2b can't ever become 0; if we plug in any number, we can make very small numbers; but we can't actually get all the way to 0.1480

We can't touch 0; the same is going to go for negative numbers.1486

If we wanted to say 2b = -4, there is no number that does that.1489

We can't raise 2 to some number and make it negative--it started out positive, so we can't possibly make it negative.1495

So, this is impossible; this is impossible; this is impossible; so it means that log2(0) is an impossible idea.1501

We can't take the logarithm of 0 or anything that is going to be negative,1509

because it just won't be possible for it to work over here, where we are trying to figure out what would be the exponential version of it.1513

So, since it just doesn't make sense to take the logarithm of a number that is 0,1519

or to take the logarithm of a number that is negative, it must be that the domain is always positive.1523

We have to go from 0, but not including 0, all the way up to positive infinity.1528

We can take in any of those things, but we can't take in 0; we can't take in negative numbers.1533

That explains why our domain has to be this.1537

We can think about it that way, or we can think about it as this flipped idea of the fact that exponentiation and logarithms are inverses.1540

So, we can have this more complex idea of the domain and range of what those things have to be.1546

But we can also just go to the fact that it does not make sense--it would be nonsense; that is a reasonable idea, too.1552

All right, we are ready for some examples.1559

Let's evaluate these numbers without a calculator.1561

If we are looking at log6(216), then that is going to be equal to some number,1564

such that, when we raise 6 to that number, we get 216.1568

216 = 6?: we want to figure out what this is, so let's see.1572

What are some numbers that we could get out of this?1577

61--that is just 6; 62--well, that would be 36; 63 is 180 + 36 = 216.1579

That is what we are looking for; so it must be the case that it is 3: log6(216) = 3; that is our answer.1588

If we have log(1/10000), the first thing we want to do is remember: if we have just log, then that is a way of saying it is log base 10.1600

So, log10 (1/10000)--once again, we are asking what that is going to be.1609

Well, that is going to be the number such that 10 to whatever that number is is going to be equal to 1/10000.1613

So, let's look at possible numbers for 10; if we go positive, we have 101 = 10.1624

Well, that is not going to work, because we are going to need a fraction.1629

So, we notice that 10-1 is 1/10; and then, if we think about that, 10-1 would be 1/10;1632

10-2 would be 1/100; 10-3 would be 1/1000; 10-4 would be 1/10000.1638

So, 10-4 = 1/10000; we can also see this, because we can count the number of 0's: 1, 2, 3, 4.1645

So, that is 104, and since it is 1/104, then that must be 10-4.1655

We have that -4 is what we have to raise 10 to, to get 1/10000.1661

The natural log of e17: well, remember: natural log is just another way of saying log base e.1668

So, loge(e17): what number do we have to raise e to?1673

e? =...well, the thing we are working with is e17, so e17 would be e?.1683

Well, that is pretty clear: the thing that the question mark has to be is the 17.1691

Otherwise they will never match up; so it must be e17 that we want here, so 17 is our answer,1695

because if we raise e to the 17, it is no surprise that we get e17.1702

Finally, log4(32): once again, we are saying, "What is the number that we have to raise 4 to, to get 32?"1707

So, we move that over; we can think of this as 4? = 32.1714

So, 32 = 4?: well, let's start looking at some possible numbers for 4.1720

We could have 41; that would just be 4--not big enough.1725

42 would be 16; we are starting to get close.1730

43 would be 64--it looks like we overshot.1733

Well, we might notice that 16 times 2 equals 32; so if we could somehow get 2 to show up, we would be good.1738

Notice: how is 4 connected to 2--what is the connection between these numbers?1746

Well, the square root of 4 is equal to 2; but we also had another way of saying that: 41/2 is the same thing as saying square root.1753

41/2 = 2; so we see that 42 times 41/2 equals 32.1762

4 squared times 4 to the 1/2 (4 squared is 16; 4 to the 1/2 is 2)--16 times 2--gets us 32.1770

So now, we just need to combine those: 42 times 41/2 is just another way of saying 44/2 times 41/2.1778

We can add them--they are on a common base; so, 45/2...1790

So, the answer for this--the number that we have to raise 4 to, to get 32, is 5/2.1795

All right, what if we were doing the other direction--if we wanted to write an exponential equation in logarithmic form?1803

We have these exponential equations: 34 = 81--and now we want to do it in the logarithm form.1809

Remember, we have that loga(x) = y is the same thing as saying ay = x.1814

Remember, our base here--we can think of it as popping up under what is on the other side of the equation.1824

So, this over here is the exponential form; this here is the logarithmic form.1830

Logarithmic form is this log stuff, and exponential is this a to the something stuff.1838

So, what we have is exponential forms here; we want to flip it.1843

34 = 81: that is going to be log...what is our base? Our base here is a 3, so log3...1847

what is the number that we are raising to? That is the blue, so we are not going to use that.1858

Finally, the number that we have is x...so, log3(81) = the number that we have to raise to, 4, because 34 = 81.1862

If we ask what number we have to raise 3 to, to get 81, that is going to be 4; 34 = 81.1878

We can do this with any of this stuff: 102.4 = 251.18.1887

Then, that is going to be...our base is 10, so we can write that as just log, because if we don't have a base, it just says log base 10.1894

log of what number? Our number that we are going to get to is 251.18,1902

and it actually keeps going, so we will leave those dots there to show that it keeps going.1908

And that is going to end up equaling 2.4, because the number that we have to raise 10 to,1915

to get 251.18, is 2.4, as was shown to us in our original exponential form.1921

So, another one: our base here is e, so we can write that as natural log of this number.1929

We were told that it comes out to be 4.1937

Finally, our base is π: so logπ(this number) = √(1/2), because we know that,1942

if we raised π to the √(1/2), we would get 2.2466, and continuing on.1955

So, that is how we were able to figure out that logπ(2.2466...) must be the square root of 1/2.1962

Graph f(x) = log3(x): to do this, we want to start with a nice table to figure out the values.1971

x; f(x); notice that we probably don't want to just toss in numbers right away.1978

If we plug in 10, well, I don't know what number we have to raise 3 to, to get 10.1986

That is going to be something complicated; we have to use a calculator.1992

But we do know what it is going to be if we plug in numbers like, say, 3.1994

If we plug in 3, what number do you have to raise 3 to--what is log3(3)?1998

What number do we have to raise 3 to, to get 3?2004

That is easy: we just have to raise it to the 1--nothing at all.2006

We don't have to raise it to anything, other than what is already there; so just something to the 1 is what it starts as.2009

What about 9?--well, what number do we have to raise 3 to, to get 9?2015

We have to square it, so we have to raise it to the 2.2019

We can keep going in this pattern: what number do we have to raise 3 to, to get 27?2021

We have to raise it to the 3.2025

What number do we have to raise 3 to, to get 81? We have to raise it to the 4.2027

And we can keep going if we want.2031

What if we went in the other direction? Well, for 2, we don't know what number we would have to raise 3 to.2032

But for 1, yes, we do know what number we would have to raise 3 to.2037

3 to the what equals 1? Just like everything else, 3 to the 0 equals 1.2039

We could go to 1/3: what number do we have to raise 3 to, to get 1/3? -1.2047

What number do we have to raise 3 to, to get 1/9? -2.2053

And it would get lower and lower and lower, the closer we got to 0.2057

Once again, we will never actually be able to get to 0, because there is no number that we could raise 3 to, to get 0.2060

But we can get really, really close to 0.2065

So, at this point, we are ready to actually try plotting it.2067

Notice: our x-values go pretty widely; so let's look at x-values going from -10 up to +100.2069

And let's look at our y-values: our y-values, our f(x) values, don't really manage to change very much.2078

So, we will look at y-values only going from -3...oh, let's make it -5...up to positive 5.2083

OK, let's start drawing that in; we start here; here is our x-axis and y-axis.2092

Make a scale; the scale for the x will be in chunks of 10, because we have to cover a lot of ground: -10, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100.2103

We can keep going if we want, but that is good enough for us.2120

So, here is a 10; here is 100; I will mark 50 in the middle; 1, 2, 3, 4, 5....50, and -10.2123

So, we can see the scale on it here.2131

For our verticals, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5.2133

Here is -1 and -5, positive 1 and positive 5.2144

Great; now we are ready to plot down some points.2150

We see a 3; we are at 1; so we are very close here; we are a little under 1/3 of the way up to the 10.2152

So, let's say it is about here; at 9, just a little bit before the 10, we are at 2.2159

At 27 (10, 20, 30--a little bit before that, but a little under 1/3 of the way towards the other side), we are at 3.2167

At 81 (100, 90, 80--just a hair in front of 80), we manage to be at 4.2180

There we go; now we want to go the other way, as well.2190

At 1/3...at 1, we are at 0, so we are really, really close to that y-axis already; at 1/3,2194

now we are getting pretty close; we are at -1 at 1/9; we are practically on top of it,2202

but we will never actually be on top of it; we will just get really, really close.2206

And we can see that this pattern is going to continue: 1/27, -3; 1/81, -4.2210

So, as it gets really close to the 0, it is going to just shoot down really quickly.2216

So, let's draw this side in; it approaches this asymptotically.2221

It gets really, really close, but it will never actually touch it; the part where it looks like it is touching it is just my human error at fault.2228

But it is not going to ever quite touch it on a perfect graph.2235

It might look like it, because of the thickness of the lines, but it will never actually do it.2239

And as it grows more and more, it slows down, because it has to go even farther out to be able to get any growth.2242

It slows down the farther out it gets; and we graph log3(x).2250

Cool; finally, what are the domains of these functions? f(x) = log7(-x + 2).2254

Remember: the idea we had was loga(stuff); then this stuff here must always be positive.2261

So, it must be positive; otherwise, it just doesn't work.2271

If we try to take the log of 0, it doesn't work; if we try to take the log of a negative number, it doesn't work.2276

You always have to take the log of positive numbers, whatever the base is.2282

For any base, this is going to be the case; so it doesn't matter if it is base 7 or base fifty billion.2286

It is going to be the case that we have to have whatever is inside of the logarithm,2290

whatever the logarithm is operating upon--it has to be greater than 0; it has to be a positive number.2294

So, we know that the thing that log is operating on here is -x + 2.2300

So, we know that -x + 2 must be positive; it must be greater than 0.2304

We move the x over; we have that 2 has to be greater than x, so x has to be less than 2,2308

and it can go all the way down to negative infinity, because the only restriction we have is that 2 is greater than x,2314

which we could write out as...anywhere from negative infinity up until positive 2, but not including positive 2,2318

which we show with a parenthesis to show that it is not inclusive.2325

Over here, g(t) = 5t(logπ(3t + 7)).2329

Once again, the base doesn't really matter; it has to be positive, no matter what the base is.2333

For any arbitrary base a, it has to be positive on what the logarithm is operating on.2339

We look at the 5t part: we might get worried--"oh, is the 5t going to interact with it?"2345

5t times logπ...5t is really in its own world; it is doing its own thing.2349

5 times t...we can do that for any number; we can multiply 5 times any number, so its domain is anything at all.2353

It is not going to actually get in our way; once again, the only thing we are worried about is2359

when the logarithm is going to try to take the log of a negative or 0 number.2362

So, to avoid that, we have to have that 3t + 7 must be greater than 0;2368

otherwise, we will be taking the log of something that we cannot take logs of, that would break our function.2373

So, 3t + 7 is greater than 0; subtract 7; 3t > -7; divide by 3; t must be greater than -7/3.2378

So, t starts at -7/3, but is not actually able to include -7/3.2388

So it starts just above -7/3 and can go anywhere larger; so it can go all the way out to positive infinity.2392

So, we have -7/3 shown with a parenthesis, because we can't actually include -7/3; we can just get arbitrarily close to it.2397

It is going all the way out to positive infinity.2404

And there are our two domains; all right, cool.2407

We will talk a bunch more about logarithms in the next one, where we will explore the properties of logarithms.2409

And then, we will see even more about how the two connect.2413

We have a lot of really interesting ideas; it is new stuff, but once you start practicing it,2414

as you do it a bunch of times, logarithms will really start to make sense.2418

You will get this idea of "what am I trying to raise this number to, to get the thing I am taking the logarithm of?"2420

What does this base have to be raised to, to get the number that I am taking the log of?2426

All right, we will see you at Educator.com later--goodbye!2430