  Vincent Selhorst-Jones

Solving Exponential and Logarithmic Equations

Slide Duration:

Section 1: 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
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
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
18:50
19:04
19:21
20:04
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
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
Section 2: 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
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
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
Section 3: 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
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
13:08
13:21
Form of Factored Binomials
13:38
Factoring Examples
14:40
16:58
Factoring Higher Degree Polynomials
18:19
Factoring a Cubic
18:32
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
9:56
11:38
Derivation
11:43
Final Form
12:23
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
22:00
Example 3: Solve for Zeroes
25:28
Example 4: Using the Quadratic Formula
30:52

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
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
26:49
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
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
19:24
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
6:29
Non-Distinct Roots
6:59
Denoting Multiplicity
7:20
7:41
8:55
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
Section 4: 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
8:25
9:26
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
Section 5: 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
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
Section 6: Trigonometric Functions
Angles

39m 5s

Intro
0:00
Degrees
0:22
Circle is 360 Degrees
0:48
Splitting a Circle
1:13
2:08
2:31
2:52
Half-Circle and Right Angle
4:00
6:24
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
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
9:43
Mnemonic: All Students Take Calculus
10:13
13:11
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
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
Section 7: 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

52m 52s

Intro
0:00
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
Section 8: 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
Section 9: 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
11:42
Example of Using a Point
12:41
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
11:14
Example 1
13:24
Example 2
15:50
Example 3
22:02
Example 4
29:06
Example 4, cont.
33:40
Section 10: 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
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
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
Section 11: 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
Section 12: 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
Section 13: 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
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
Section 14: 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
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
11:51
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
Section 15: 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
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
Section 16: 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
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
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
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
Section 17: Appendix: Graphing Calculators

10m 41s

Intro
0:00
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
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
5:44
Which Graphing Calculator is Best?
5:46
Too Many Factors
5:54
6:12
The Old Standby
7:10
TI-83 (Plus)
7:16
TI-84 (Plus)
7:18
9:17
9:19
9:35
10:09
Graphing Calculator Basics

10m 51s

Intro
0:00
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
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
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
5:10
Find Out What Input Value Causes a Certain Output
5:12
For Example
5:24
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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• ## Related Books 0 answersPost by Angela Zhou on March 26, 2018For example 3, is there a mistake? For log5 11, you can’t multiply this by x 1 answer Last reply by: Professor Selhorst-JonesFri May 13, 2016 7:44 PMPost by Raoul Benjamin on May 13, 2016for Example 4:how come it does not work if you write 12 as 4^1.79248 then cancel out the 4's and just solve for t? 2 answersLast reply by: Napolean RichardMon Jan 19, 2015 10:36 AMPost by Napolean Richard on January 6, 2015Dear sir,In this practise quesetion:lnx + ln(x+3) = 2 .Why could not i make both side an e equation exponent to cancel out ln directly?Then the next is x+(x+3)=e^2? 0 answersPost by Saadman Elman on November 11, 2014The concept became very clear now. Thanks!

### Solving Exponential and Logarithmic Equations

• Now that we understand how exponents and logarithms work, how do we solve equations involving them? There are two main ideas that allow us solve such equations: the one-to-one property and the inverse property.
• The one-to-one property points out that exponential and logarithmic functions are both one-to-one, that is, different inputs always produce different outputs.
 ax = ay   ⇔   x = y                      logb x = logb y   ⇔  x=y
If an equation is set up in one of the formats above, we can turn it into something fairly easy to solve.
• The inverse property points out that exponentiation and logarithms are inverses of each other: if they have the same base, they "cancel out".
 loga (ax) = x                             blogb x = x
This means we can make both sides of an equation exponents or take the logarithm of both sides to "cancel out" things that are "in the way" of solving an equation.
• One particularly useful property of logarithms is
 loga xn   =  n ·loga x.
This allows us to "bring down" exponents with any logarithm. Furthermore, we can use logarithm bases that we have on our calculators to make calculation easy (e⇒ ln,  10 ⇒ log).
• Many of the properties we've discussed about exponents and logarithms in previous lessons can be useful in solving exponential or logarithmic equations. If a problem is complicated, try to figure out if you can first simplify it with some of the various properties we've learned.
• While solving these equations, it's important to watch out for extraneous solutions: values that appear over the course of solving, but are not actually solutions. A value might seem like a solution, but actually be outside of the allowed domain. Make sure to check your answers to be sure they do not cause any issues.

### Solving Exponential and Logarithmic Equations

Solve for x.
 472x+9 = 473x−1
• We begin by using the one-to-one property. The only way this expression could be true is if the two exponents are equal to each other. This means we can set them equal in an equation, then solve.
• From the one-to-one property, we have
 2x+9 = 3x−1,
which we can solve for x using basic algebra.
x=10
Solve for t.
 ln(t+17) = ln(6t+2)
• We begin by using the one-to-one property. The only way this expression could be true is if the two quantities that the logarithm acts upon are equal to each other. This means we can set them equal in an equation, then solve.
• From the one-to-one property, we have
 t+17=6t+2,
which we can solve for t using basic algebra.
t=3
Solve for a.
 36a+3 = 6−4a − 6
• Notice that we can't immediately use the one-to-one property because we don't have the same base for the exponent on either side of the equation. However, notice that 36 and 6 are connected through exponentiation, so we can find a way to rewrite one of the sides to match the other side. [We could also approach this problem by taking the log of both sides, bringing down the exponents, then working through it, but that makes things a little more difficult (and could potentially give us an inaccurate answer if we allow any mistakes to creep in).]
• We see that 36=62, so we can substitute that in:
 36a+3 = 6−4a − 6    ⇒     (62)a+3 = 6−4a − 6     ⇒     62(a+3) = 6−4a − 6
• Now that either side has the same base for the exponent, we can make an equation out of the exponents using the one-to-one property:
 2(a+3) = −4a − 6
At this point we can solve for a using basic algebra.
a=−2
Solve for x exactly.
 10−7x+3 = 1000
• The problem here is that our variable is "stuck" in an exponent. We can get it out of the exponent by using an appropriate logarithm. This is called the inverse property:
 loga (au) = u                             blogb v = v
• By taking log of both sides (remember, log with no number is short-hand for log10), we can "free" the exponent:
 10−7x+3 = 1000     ⇒     log( 10−7x+3 ) = log1000     ⇒     −7x + 3 = log1000
• Furthermore, we can use a calculator to find that log1000 = 3 (or we can realize that 1000 = 103 and get the same result), thus we have:
 −7x + 3 = log1000     ⇒     −7x + 3 = 3     ⇒     x=0
• As an alternative method to doing this problem, we could also realize from the beginning that 1000=103, allowing us to make the substitution
 10−7x+3 = 1000     ⇒     10−7x+3 = 103
and then use the one-to-one property to see that
 −7x+3 = 3.
Either way of approaching the problem will work, but it's good to start seeing that logarithms can be used to "cancel out" the bases of exponents we want to work with.
x=0
Solve for x exactly.
 log8 x5 − 2 log8 x3 = log8 x4 − 10
• While we could attempt to put each side as an exponent on a base of 8 to cancel out the log8's, that would get messy. It would be possible, but it would be a little bit of a pain to work with. What we can do instead is try to combine the logarithms as much as possible through the use of logarithm properties and simple algebra before we try to cancel out the log8.
• By the properties of logarithms, we can bring down the exponents-loga xn = n ·loga x:
 log8 x5 − 2 log8 x3 = log8 x4 − 10     ⇒     5log8 x − 6log8 x = 4log8 x − 10
• Notice that we can now simplify things by combining the like terms of log8 x:
 5log8 x − 6log8 x = 4log8 x − 10    ⇒     −1log8 x = 4log8 x − 10
Then, we can manipulate through basic algebra, adding log8 x + 10 to both sides:
 −1log8 x = 4log8 x − 10     ⇒     10 = 5 log8 x
• Continue to simplify a bit more:
 10 = 5 log8 x     ⇒     2 = log8 x
At this point, we can get rid of the log8 by raising both sides up as exponents with a base of 8:
 82 = 8log8 x     ⇒     64 = x
If you're not sure how the above step worked, remember that logarithm and exponentiation of the same base cancel each other out. This is called the inverse property:
 loga (au) = u                             blogb v = v
x=64
Solve for p below. Name the extraneous solution that you find while solving.
 log3 (p2−5) = log3 (5 − 3p)
• Since we have the same log on both sides and nothing else, things are conveniently set up to use the one-to-one property:
 log3 (p2−5) = log3 (5 − 3p)    ⇒     p2−5 = 5−3p
• Since we have a polynomial structure, we solve by getting 0 on one side and factoring:
 p2−5 = 5−3p     ⇒     p2 + 3p −10 = 0     ⇒     (p+5)(p−2) = 0
Thus, solving the polynomial gives us two values for p: −5 and 2.
• However, just because the values are solutions to the polynomial equation does not necessarily mean they are solutions to the logarithmic equation we started with. Remember, for a logarithm to work, it must operate on a number in the set (0, ∞). Thus, we need to check and make sure that plugging in the number won't "break" the logarithm.
 p=−5:       log3 ⎛⎝ (−5)2−5 ⎞⎠ = log3 ⎛⎝ 5 − 3(−5) ⎞⎠ ⇒      log3(20) = log3 (20)

 p=2:       log3 ⎛⎝ 22−5 ⎞⎠ = log3 ⎛⎝ 5 − 3(2) ⎞⎠ ⇒      log3(−1) = log3 (−1)    BAD
Because p=2 causes a negative to appear in the logarithm (thus "breaking" it), we must discard it as a solution. We call solutions that initially appear as possibilities but are later discarded extraneous solutions.
p=−5;        Extraneous solution: p=2 [Note that p=−5 is the only correct solution to the problem, p=2 is just a value that we initially thought would work, but are later forced to throw out as extraneous.]
Solve for x exactly, then also give an approximation to four decimal places.
 4ex = 27
• We can't use the one-to-one property here because we have no corresponding e base on the other side. This means we will have to solve this by using the inverse property to cancel out the e base of ex. We will do this by taking the natural log (ln) of both sides. [Remember, ln is just short-hand for loge.]
• To make it a little easier for us, begin by solving for ex on one side of the equation:
 4ex = 27     ⇒     ex = 27 4
• Now we can take the natural log of both sides so that we can cancel out the e base and get x alone on one side:
 ex = 27 4 ⇒     ln(ex ) = ln ⎛⎝ 27 4 ⎞⎠ ⇒     x = ln ⎛⎝ 27 4 ⎞⎠
At this point, we have solved for x, and we see it is precisely the value of ln([27/4]). We cannot simplify it any more. [This is because the natural log of most numbers is like the square root of most numbers. While there are a very few numbers that will simplify nicely (e, e2, e3, ek, etc.), most numbers cannot be simplified any further. We can write out the precise value using the logarithm, or we can plug it in to a calculator to get a decimal approximation.
• Plugging into a calculator, we find
 x = ln ⎛⎝ 27 4 ⎞⎠ ≈ 1.9095
x = ln( [27/4] ) ≈ 1.9095
Solve for t.
 logt3 + logt2 = 5
• We will want to use the inverse property to cancel out the logarithm, but it will make our work easier to begin by combining the logarithms into a single thing before canceling out. We can do so with the logarithm property loga (M ·N) = loga M + loga N then simplifying:
 logt3 + logt2 = 5     ⇒     log(t3 ·t2) = 5     ⇒     log(t5) = 5
• We can make things even easier by "pulling down" the exponent in the logarithm by using the property loga xn = n ·loga x:
 log(t5) = 5    ⇒     5 logt = 5
Then, using basic algebra to divide by 5 on both sides, we get:
 5 logt = 5     ⇒     logt = 1
• Now that it's in a nice, simple form with the logarithm alone on one side, it is easy to use the inverse property to cancel out the logarithm. If we put both sides as exponents with a base of 10, we can cancel out the logarithm (remember, log is short-hand for log10):
 logt = 1     ⇒     10logt = 101     ⇒     t = 10
t=10
Solve for z precisely, then give an approximation to four decimal places.
 4 · ⎛⎝ 1 3 ⎞⎠ 2z+3 = 8
• Begin by isolating the base/exponent combination:
 4 · ⎛⎝ 1 3 ⎞⎠ 2z+3 = 8     ⇒ ⎛⎝ 1 3 ⎞⎠ 2z+3 = 2
• Next, we want to "get access" to the (2z+3) exponent on the base. At this point, there are two options: first, we could take log[1/3] of both sides. This would cancel out the [1/3] base, but has the downside that very few calculators can tell us what log[1/3] 8 is. [Although we could use the change of base formula later on.] Alternatively, we can use a base that most calculators can easily work with: 10 or e. Either works fine, so let's arbitrarily pick log to work with. While log can't cancel the [1/3] (because it has a base of 10, which is different), we can use one of the logarithm properties that allows us to "pull down" the exponent-loga xn = n ·loga x:
 ⎛⎝ 1 3 ⎞⎠ 2z+3 = 2     ⇒     log ⎡⎣ ⎛⎝ 1 3 ⎞⎠ 2z+3 ⎤⎦ = log2     ⇒     (2z+3) log ⎛⎝ 1 3 ⎞⎠ = log2
Using this method will allow us to easily find the decimal approximation with our calculators at the end.
• Now we can go about solving for z like normal with algebra. Just be careful to treat the logarithms as quantities while working.
(2z+3) log
1

3

= log2     ⇒     2z + 3 = log2

 log 1 3
⇒     z = log2

 2 log 1 3
3

2
• We can't simplify things any further, so now we use a calculator to approximate the value:
z = log2

 2 log 1 3
3

2
≈ −1.8155
• In case you're curious about how we could have done the first method (using log[1/3]), here's a quick run-down. First, cancel using log[1/3]:
 ⎛⎝ 1 3 ⎞⎠ 2z+3 = 2     ⇒     log[1/3] ⎡⎣ ⎛⎝ 1 3 ⎞⎠ 2z+3 ⎤⎦ = log[1/3] 2     ⇒     2z + 3 = log[1/3] 2
Then solve for z:
 2z + 3 = log[1/3] 2     ⇒     2z = log[1/3] 2 − 3     ⇒     z = log[1/3] 2 − 3 2
This is an equivalent way to give z, the only problem is that most calculators can't directly give us a decimal value because of the logarithm base. Use the change of base formula to obtain
log[1/3] 2 = log2

 log 1 3
,
then plug that in when evaluating z to find the decimal approximation.
z = [ log2/(2 log[1/3] )] − [3/2] ≈ −1.8155 [Alternatively, you could give the equivalent answer of z = [(log[1/3] 2 − 3)/2]. Both are correct, just two different ways of getting to the solution. See the steps for more of an explanation.]
Solve for x precisely, then give an approximation to four decimal places.
 lnx + ln(x+3) = 2
• Begin by assembling your logarithms into a single logarithm so that it will be easier to cancel the logarithm out:
 lnx + ln(x+3) = 2    ⇒     ln ⎛⎝ x(x+3) ⎞⎠ = 2     ⇒     ln(x2 + 3x ) = 2
• Using the inverse property, cancel out the logarithm by raising both sides as powers:
 ln(x2 + 3x ) = 2     ⇒     eln(x2 + 3x ) = e2     ⇒     x2 + 3x = e2
• Now we need to solve for x. Notice that e2 is just a number, so we've got a polynomial equation. While e2 is too complicated for us to solve by factoring, we can use the quadratic formula, so begin by getting everything on one side:
 x2 + 3x = e2    ⇒     x2 + 3x − e2 = 0
Now we can set up the quadratic formula. We have a = 1, b=3, and c=−e2:
x = −b ± √ b2−4ac

2a
=     −3 ± √ 32 − (4)(1)(−e2)

2·1
• Simplify:
 −3 ± √ 32 − (4)(1)(−e2)

2·1
=     −3 ± √ 9+4e2

2
We can't simplify any more at this point, so we see we now have two solutions from solving the polynomial equation:
x = −3 + √ 9+4e2

2
≈ 1.6047            x = −3 − √ 9+4e2

2
≈ −4.6047
• However, just because those are the solutions to the polynomial equation does not necessarily mean they will satisfy the original, logarithmic equation. Remember, logarithms can only operate on numbers greater than 0. If we were to plug in the second "solution", we would get
x = −3 − √ 9+4e2

2
: ⇒ ln(−4.6047) + ln(−4.6047 + 3) = 2     BAD.
Since we can't take the logarithm of a negative number, we must throw that out as an extraneous solution.
• While we're at it, it's probably a good idea to check that the other solution does, in fact, work. After a tough problem, it is a good idea to check your answer to make sure there aren't any mistakes We can do this by plugging in our approximate value and using a calculator. Plugging in for the left side, we have
x = −3 + √ 9+4e2

2
: ⇒ ln(1.6047) + ln(1.6047+3)     =     2.000  014…
Since the right-hand side of the original equation is 2, this checks out. [The reason it did not come out precisely to 2 is because of rounding error. We got the value of 1.6047 only after rounding to four decimal places, which introduced a slight error.]
x = [(−3 + √{9+4e2})/2] ≈ 1.6047 [Note: The above is the only solution. If you got two values for x, don't forget to check for extraneous solutions.]

*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.

### Solving Exponential and Logarithmic Equations

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 0:05
• One to One Property 1:09
• Exponential
• Logarithmic
• Specific Considerations
• One-to-One Property
• Solving by One-to-One 4:11
• Inverse Property 6:09
• Solving by Inverses 7:25
• Dealing with Equations
• Example of Taking an Exponent or Logarithm of an Equation
• A Useful Property 11:57
• Bring Down Exponents
• Try to Simplify
• Extraneous Solutions 13:45
• Example 1 16:37
• Example 2 19:39
• Example 3 21:37
• Example 4 26:45
• Example 5 29:37

### Transcription: Solving Exponential and Logarithmic Equations

Hi--welcome back to Educator.com.0000

Today, we are going to talk about solving exponential and logarithmic equations.0002

At this point, we have a good understanding of how both exponentiation and logarithms work.0006

However, we haven't seen much about how to solve equations involving them.0010

For example, how do we solve something like e2x - 3 = e5x - 12? Or log(x - 2) = 1?0013

We haven't really talked about how to do that.0023

We briefly touched on it in the final example in the last lesson, but now we are going to really explore it.0024

We will go over two different ways of approaching such equations.0029

First, we will discuss how we can use the 1:1 property, and then we will see how we can apply the inverse property for more complicated equations.0032

Make sure you already understand how exponents and logarithms work.0040

The previous few lessons explain this stuff in detail--explain how all of these things work and their properties.0044

So, make sure you get an understanding of that before you try to get into these equations.0049

While you might be able to understand it, it will make a lot more sense if you already have a grasp0052

of how exponents and logarithms work and what their properties are.0057

Then, you will really be able to throw yourself into the equations.0060

If you don't have a good grasp of that, first make sure that you have watched those lessons previously,0062

because they will really help out for understanding this lesson.0065

All right, notice that the exponential and logarithmic equations are both one-to-one: different inputs imply different outputs.0069

If we put a different value into the function, a different output always will come out.0076

We can see this in the graphs, because both function types pass the horizontal line test.0081

If we cut an exponential function's graph with a horizontal line, it is only going to intersect at a maximum of one point.0085

This means that it is a one-to-one function.0093

If you want a better understanding of one-to-one functions, you want to check out the Inverse Functions lesson.0097

The same with logarithmic functions: if we cut at any place with a horizontal line, it will only intersect one time.0104

Thus, they are one-to-one functions: for any input, there is only one output; for every output that can come out of it, there is a unique input that creates it.0111

As an example, let's consider some specific numbers: if we wanted to see 34 = 3 to the something,0123

what could go into that box to give us 34?0129

Well, if we think about it for a while, we will probably think, "Well, there is only one choice to go into that box."0132

It doesn't make sense for there to be anything else; the only other thing that could be on the other side of the equation is also 34.0137

The only thing that is going to make 34 with a base of 3 is 34.0143

So, the only thing that can go inside of this box is a 4.0148

Similarly, over here with log5(25) = log5(something),0152

the only thing that could possibly be in there...if we think about this for a while, we will see,0157

"Well, the only thing that could make any sense to go inside of that box is a 25."0160

It wouldn't work any other way; there is nothing else that we could take log base 5 of and get the same value as log5(25).0165

So, the only thing that could go in those boxes is the number on the other side of the equation.0174

This is because they are one-to-one--this one-to-one property.0178

Since we have an input of 4 over here, and it gives us an output of 34, 81,0181

we know that the only other input that is going to be able to do that is no other input.0187

This 4 is the only input that can give us 81 as our output; so the same thing must be here--it is one-to-one.0192

If an input gives an output, the only input that can give that output is that original input.0199

There is no other input; it is unique to the output.0205

We call this the one-to-one property; it says that, if the base of a is raised to x, and we have that equal to the same base, a,0211

raised to y, then it must be the case that x equals y.0219

Similarly, if we have logb acting on x, and we know that it is equal to logb acting on y, then we know that x = y.0223

And notice that, in both of these cases, the base has to be the same on each one.0231

We have ax and ay; we have logb and logb--0235

logb(x) and logb(y)--we have the same bases in both of these cases.0241

And that is why we have this one-to-one property working: we know that x has to equal y in both of these situations.0246

All right, with this property in mind, we can now solve equations where we have an exponent or a logarithm of a single base on both sides of the equation.0252

For example, if we have e2x - 3 and e5x - 12, well, since we have the same base on both sides,0259

we know that 2x - 3 has to be the same thing as 5x - 12.0266

Otherwise, we wouldn't have equality.0270

So, we know that 2x - 3 = 5x - 12; and at this point, we go about it just like we are solving a normal algebra equation.0271

Add 12 to both sides; we get 2x + 9 = 5x; subtract 2x from both sides; we get 9 = 3x; divide by 3 on both sides; we get 3 = x.0279

If we want to check it, we can plug in our 3 = x: e2(3) - 3 = e5(3) - 12.0291

So, we have e6 - 3 = e15 - 12, so e3 = e3; and that checks out just fine.0304

The same basic idea over here: log7(x + 5)...and let's move the other log7(2x - 3).0315

We will add log7(2x - 3) to both sides; so it will now appear on the right side.0322

At this point, we know that what is inside of both logs has to be the same thing.0331

They are both log7, so we know that they must be taking logarithms of the same object.0336

x + 5 and 2x - 3 must be the same thing; otherwise, we couldn't have equality.0344

We have x + 5 = 2x - 3 by this one-to-one property.0349

We can add 3 on both sides; we will get 8; so x + 8 = 2x.0355

We can subtract by x on both sides; we will get 8 = x; and there is our answer.0362

And if we wanted to, we could check that one the same way.0367

The inverse property: previously, we have talked about exponentiation and logarithm, if they have the same base, are inverse processes.0370

If they are applied on after another, they cancel each other out.0378

So, if we have natural log on e2, well, since natural log is just the same thing as log base e,0383

and it is operating on something that is base e, to the 2, then the log base e here and the e cancel out, and we are left with just 2.0390

So, sure enough, 2 = 2; that is the idea of what is going on there.0397

The same thing over here: 5 as our base, raised to the log5(125)--they cancel out, and we get 125.0401

That is how this stuff is coming out.0409

If you want a more in-depth exploration of this, check out the previous lesson, Properties of Logarithms,0410

where we will actually prove this stuff and see why it has to be the case.0414

We call this the inverse property: so loga(ax) is equal to x.0418

And blogb(x) = x; so we have this cancellation.0424

If we end up having the same base like this in log base, and then exponent base, or in exponent base and then log base,0429

we will cancel out, and we will just get the thing that is at the end.0437

So, in this case, that is x here or x here that we end up getting out of it.0440

Solving by inverses: with an equation or inequality, we can do algebra.0446

Now, algebra is just applying the same thing to both sides; we are doing the same operation--whatever it is.0451

When we first learned to do algebra, we just used simple arithmetic (things like addition, subtraction, multiplication, or division).0458

what we just saw in the previous thing when we were solving after we used these special properties.0466

But as we learned more in our work in algebra, we realized that we could do more than just apply simple operations on both sides.0470

We could even do more complicated things, like squaring both sides, or taking the square root.0477

We realized that, as long as we are doing the same action to the left side and the right side--0482

we are doing the same thing to two things that are equal--0486

we know that it is going to remain being equal, even after the action goes through.0489

So, by that same idea, we can also make both sides of an equation exponents, or take the logarithm of both sides,0493

because we are doing the same action to both sides--the equality is still based in it.0499

If we have wavy stuff equals loopy stuff, then we know that a, as a base for wavy stuff, is going to be equal to a as a base for loopy stuff,0504

because a to the something-on-the-left equals a to the something-on-the-right, and we were told that left side and right side are the same.0515

So, it must be the same still, even when they are working as bases for something else,0521

or when something is a base underneath them and they are now exponents.0525

A similar idea: if we were told that wavy equals loopy, then we know that logb(wavy) must be the same thing as logb(loopy).0528

We are applying the same action, whether it is turning them into exponents on some base,0536

or we are taking the logarithm of some base of both sides.0540

We are doing the same action to both sides, so we have this equality still holding.0543

Combining this idea with the inverse property allows us to get rid of exponent bases or logarithms that are in the way of solving an equation.0547

For example, if we have log(x - 2) = 1 (remember, if it is just log, then that is the common log, so it is log base 10 of x - 2 equals 1);0556

what we can do is come along and raise both sides with a 10 underneath them.0566

So, we are not raising to a power, like squaring them; we are actually causing this exponent base to erupt underneath them.0572

We have 10 to the log10(x - 2); and that is going to be equal to...10 has erupted underneath the 1.0579

All right, at this point, we have the inverse property; we are solving by inverses, so we have 10 and log10.0589

So, these cancel out, and the x - 2 just drops down; and we get x - 2 =...10 to the 1 is just 10.0595

So now, we just solve it normally: x = 12--there is our answer.0603

A similar idea is going on over here: 32x = 7--well, let's get rid of that base of 3; that is getting in our way.0608

So, we will take log3(32x), and that is going to be equal to log3 of what is on the right side, so log3(7).0615

log3(32x)...those will end up canceling out, and the 2x will just drop down;0627

so we will have plain 2x = log3(7).0634

Now, if we want to figure this out with a calculator...0639

log3(7) is still correct, but if we want figure it out with a calculator--if we actually want a decimal version--0641

we will have to turn it through change of base; let's take natural log--I like natural log.0647

ln(7)/ln(3): remember the change of base formula that we talked about in the previous lesson.0653

x =...now we are dividing by 2 on both sides, so it will show up in the denominator: 2 times the natural log of 3,0659

which will end up working out to approximately 0.8856.0666

Now, notice: 0.8856 is approximate; it is not the exact answer.0672

This is actually the exact answer; once we have calculated through with ln(7) and ln(3), we end up getting something0677

that is very, very close, but it is no longer precise, because we are having to cut off some of the decimal places.0684

And if we wanted to, we also could have used any other base.0689

We could have used log10(7)/log10(3)--a common log there.0691

We would have had log(7)/2log(3), which would end up coming out to be the same thing when we used a calculator on it.0699

And this would also be just equally as correct an answer.0713

A useful property: one particularly useful property of logarithms is this ability to bring down things.0718

If we have loga(xn), then that is the same thing as if the n had been in the front, if it was n times loga(x).0724

So, we can bring down exponents with any logarithm.0731

This means that we can use logarithm bases that we have on our calculators.0735

That might be convenient for us sometimes.0738

So remember: e is the same thing as natural log; 10 is the same thing as log without a number on it.0740

So, if we want, we could just start by taking the natural log of 32x = the natural log of 7.0745

At this point, we can bring down the 2x; it comes down in front by this property up here.0752

So, we have 2xln(3) = ln(7), so 2x = ln(7)/ln(3), or x = ln(7)/2ln(3), which is the exact same thing that we just had on the previous slide.0758

The thing to notice here is that we have two different ways of doing this.0780

We could go about it by using the change of base, or we could go about it by just using this property where we can bring down exponents.0783

Sometimes it will be more useful to use the bringing down the exponent property.0789

Sometimes it will be more useful to do the change of base.0792

They will both end up working out; it is just that sometimes one will be a little bit more work than the other.0794

And you will get a feel for which one you want to use as you work on these things.0798

Many of the properties we have discussed about exponents and logarithms can be useful in solving exponential or logarithmic equations.0802

If the problem is complicated, try to figure out if you can first simplify it with some of the various properties we have learned.0808

We have learned a lot of properties by this point, about how exponentiation works and about how logarithms work.0813

And sometimes, by combining things or breaking things apart, you will make the problem easier to do.0818

And we will see some of that in the examples later on.0823

Extraneous solutions: while solving these equations, it is important to watch out for extraneous solutions.0826

An extraneous solution is a value that appears over the course of solving, but isn't actually a solution--0832

that, if we were to try to use it, would just fail or cause our equation to break apart or not work or not be defined for some reason.0837

The easiest way to see how this works is to just see an example.0844

So, let's look at this: we have the natural log of x2 - 2 equals natural log of x.0847

By the one-to-one property, we see that x2 - 2 has to be equal to x.0851

Alternatively, if we wanted, we could put e's underneath it, and just cancel out both of them.0855

Inverse or one-to-one property--both end up working as ways to look at this.0858

x2 - 2 = x: at this point, it looks like the polynomials we are used to solving from that section.0862

We move it over: x2 - x - 2 = 0; so we can factor that.0870

We get (x - 2)(x + 1) = 0; we solve both of those, so x - 2 = 0; x + 1 = 0; we get x = 2 and x = -1.0875

Now, if we go back and try to work this out, if we plug in x = 2, things are pretty reasonable.0890

We get ln(22 - 2) = ln(2); so this is ln(4 - 2) = ln(2); and then, ln(2) = ln(2).0895

That is perfectly reasonable; but if we try x = -1, we will see some problems very quickly.0910

ln((-1)2 - 2) = ln(-1); and as soon as we see this right here, we get suspicious,0917

because what is the problem here? You can never take the logarithm of a negative number.0928

So, as soon as we see ln of a negative number inside, we know that this is not possible.0933

We can't take the natural log of -1; we can't have logs of negative numbers at any point showing up.0938

This is an extraneous solution--it is something that appeared over the course of solving, because we turned it into this quadratic form.0943

And in the quadratic form, it was a solution; but up here, in the original form that we have, it fails to be a solution.0952

It can't be a solution, because we end up having this logarithm of a negative number; so we knock it out--it is an extraneous solution.0960

And our only answer is x = 2.0967

So, it seems at first as though we have two answers, because we are solving a quadratic.0970

But as we work our way through the quadratic, we realize that if we were to actually plug this in0974

and try it out to see if it works as a solution, it would cause the whole thing to blow apart.0978

So, we end up seeing that it can't actually be used as a solution; so it is called an extraneous solution--0982

something that appears over the course of solving, but can't actually be used as a solution.0987

So, we have only one out of this, even though at first it seemed as though there would be two.0993

All right, we are ready for some examples now.0998

The first one: Solve for x exactly if we have 3/7x + 2 and (49/9)x - 2.1001

So, we look at this, and we think, "Well, we could take logarithms of both sides; we could bring down our exponents."1008

But things are going to get pretty messy; we will have to actually figure out what the logs are of 49/9 and 3/7,1014

and we will have to work out a bunch of numbers.1021

It is going to get really, really ugly: we could work it out that way, but we would end up having approximations,1023

because we would be taking the logs of these numbers, and they would come out to be decimals.1027

So, that won't end up working in the end.1030

But if we look at it, we might realize that 49/9...there is a connection to 3/7.1032

Well, if we want, we could rewrite this as 9/49 to the -1.1039

And then, we might realize that 9 is just 32; 49 is just 72; it is still all to the -1.1045

We can pull out the 2's, and we have (3/7)-2.1052

That is what we started with on the left side; so we can use that one-to-one property.1057

So, we take this fact here; we can swap them out; so the same thing is still on the left side, (3/7)x + 2,1062

is equal to...we swap out 49/9 for (3/7)-2, so we have ((3/7)-2)x - 2.1070

Well, that is going to be equal to (3/7)...we can bring that -2 out by the property of exponents.1083

It will multiply everything that is already out there, so we have to have that and the quantity as well.1090

So, we have (3/7)x + 2; at this point, we can use the one-to-one property, because we have the same base here and here.1094

We have (x + 2) = -2(x - 2); x + 2 = -2x + 4; add 2x to both sides; we get 3x; subtract by 2 on both sides, and we get 2,1102

and we get 3...sorry, we now divide by 3; we don't divide by 2--that would be going the wrong way.1116

Divide by 3 on both sides; we get x = 2/3; and there is our answer for x.1122

And if we wanted to, we could plug that in and check--use our calculator and end up working it out.1128

We would get decimal answers that would end up being the same thing; we would see that that ended up working.1131

So, you can check this by using a calculator if you want to; you could do a check, and you would have (3/7)2/3 + 2 = (49/9)2/3 - 2.1135

You would have to use a calculator to work this out, but if you did, you would get decimal answers that were very, very, very close--1151

actually, they should be exactly the same, because we solved for x exactly.1157

The only problem might be if your calculator has just a little bit of sloppiness in it.1160

But they should get decimal answers that are well within 5 or 10 decimal digits of each other.1164

And so, you will end up seeing that it checks out when you use your calculator.1169

Or you could also just work through each of these, and then use the properties of exponents.1173

And you could see that it is exactly the same thing--there are two ways to do it.1177

Solve for a exactly if we have log(a3) - log(a2) = 2 - log(a).1181

All right, let's use the properties of logarithms to bring some things together and simplify things a bit.1187

Remember: we have subtraction here--subtraction of logarithms is the same thing as division inside of the logarithm.1191

So, we have a3/a2 = 2 - log(a) (not base a).1198

Now, the only issue we would have is...what if a was equal to 0?1206

Well, if a was equal to 0, we would already have problems, because we would be taking the log of 0.1210

So, we don't have to worry about that; so we know that a is not equal to 0, so we are good there.1214

We know that a is not equal to 0, so we can do this cancellation.1219

We don't have to worry about that..1223

log(a3/a2)...well, we will get just log(a), because a3/a2 will cancel two of the a's on top.1224

We will be left with just one.1231

2 - log(a): at this point, we can add log(a) to both sides, so we will get log(a).1233

And now, there are 2 of them, because we added, and they are of the same type; so log(a) + log(a) is 2log(a) = 2.1238

At this point, we can divide by 2 on both sides, and we get log(a) = 1.1246

We want to know what that is: well, remember, log is just common logarithm, so it is base 10.1251

So, we can raise both sides to the 10; so that cancels this out, and we have a = 101, which means a = 10.1256

Great; if we wanted to, we can check this; so as a check, we have log(103) - log(102 = 2 - log(10).1266

log(103) comes out to be 3, because it is base 10, so what do you have to raise 10 to, to get 103?1278

Well, you have to raise it to 3; the same idea is over here--log(102) is just 2, equals 2 - log(10).1285

log base 10 of 10 is just going to be 1; 1 = 1; great--that checks out.1292

All right, the third example: Solve for x to four decimal places: 5 x + 4 = 112x.1297

So, we could write this log5 acting on 5x + 4, and then log5 on the other side, as well, acting on 112x.1304

So, since we have log5 and exponential base 5, they cancel out, and we have x + 4 on the left side.1318

On the right side, we see that we have 2x raised to an exponent; so if we want, we can bring that out to the front.1327

We have equals 2x times, and then our remaining log5(11).1333

All right, so at this point, we can divide by 2x on both sides, because we want to try to get our x's together.1341

Or actually, better yet, we can subtract x on this side, and we will get 4 = 2x(log5(11)) - x.1348

Now, at this point, we can see that there is an x here, and there is an x here; we can pull out the x's.1369

And we will get 4; pull out the x's; x times 2 times log5(11) - 1.1373

We can divide this over, so we have x =...dividing over, we have 4 divided by what we are dividing over, 2log5(11) - 1.1385

We can use the change of base formula, x = 4 over (because we probably wouldn't be able to use a calculator,1399

and lots of calculators can't do log base 5) 2 times log5(11)...let's go with natural log,1406

just because I like natural log; so ln(11)/ln(5) - 1.1413

Now, that one looks kind of ugly, and it is; it is going to take some work through a calculator.1421

But you work it through with a calculator, and you will get that that is approximately equal to 2.0204 once you round it down.1424

So, there you are: another way to have done this would have been to take the natural log of both sides.1435

We could have taken ln(5x + 4) = ln(112x), and this would be true with any base.1440

We could be doing this with the common log, as well, if we wanted to.1449

So, we can bring down our exponents; we will get (x + 4), remember, as a quantity, because it is the whole exponent,1451

times the natural log of 5, equals 2x times the natural log of 11.1457

At this point, we could move the natural log of 5 over, and we would have (x + 4) = 2x[ln(11)/ln(5)].1462

It is over the whole thing, but we can also just compact it into that one thing.1476

And then, if we want, we could move the x over as well, and we would have 4 = 2x[ln(11)/ln(5)] - x.1479

Now, we do the same trick and pull out our x's: we get 4 = x times 2 times ln(11)/ln(5) - 1.1490

We divide that over, and we get 4 over 2 times ln(11)/ln(5) - 1 = x, which is the exact same thing that we had when we did it by using log base 5.1501

So really, it is just a question of if we are prolonging the change of base or causing the change of base to happen as we take a log in a different base.1515

So, there are two different ways to do it.1523

If you don't end up realizing this x trick, where the fact that we have x here and we have x here1525

means that we can pull them both out to the front and get x at the front, you can also just work this out1530

through a bit more arithmetic and working at it, moving things around.1535

You can eventually get it; you could also just do it by evaluating what is log5(11).1540

You could figure out what log5(11) is, get that down to 8 decimal places, and then multiply that be 2x.1546

And you could work it out--just work with a whole bunch of decimal places for a while and solve for what x is,1552

and then just cut it down to four decimal places--that is another way to do it.1557

There are a bunch of different ways that you can approach problems like this.1560

Just remember all of the properties that we have talked about, and just work through it and pay attention to what you are doing.1563

And then, at the very end, check your work--do a quick check.1568

You can do a check that 52.0204 + 4 = 112(2.0204).1571

And in fact, you will end up finding out that they come out to be very, very close.1583

They will end up being different after the fifth or sixth digit, because there was a little bit of rounding error.1587

We did, after all, only round to four decimal places, and it just keeps going forever.1592

But beyond that little bit of rounding error, after the fifth or sixth digit, you will end up being really, really close to being exactly the same.1596

So, you will see that it checks out--that you do have the right answer.1603

The fourth example: Solve for t exactly--we are back to solving for t exactly,1606

so we can't really use these numerical methods that we have been working out with a calculator.1611

We need to figure out something clever here.1615

The problem here is that we have 2t and t; it is not just the same thing happening up there.1617

And then, we also have this confusion from the 12.1623

If the 12 weren't there, we could just move them over, and we could use the one-to-one property.1625

But we have this problem where we have this 42t - 4t = 12...well, let's try moving things around and see if this looks like something we are used to.1629

4t - 12 = 0; at this point, we might have a moment of understanding.1639

We might realize that this looks a lot like a polynomial--like a quadratic polynomial that we are trying to solve.1645

We realize that this is squared; this is to the one; and this is to the nothing--this is the constant.1654

So, we realize that 4t is kind of like the x that we are used to; so let's say 4t = u.1659

We will do this as a u substitution, where we will replace something complicated with something simple, this nice u.1665

If it is 42t, it is now going to be u2,1671

because u2 would be (4t)2, which is the same thing as 42t.1674

Minus u minus 12 equals 0--at this point, it is really easy to solve.1680

We see (u - 4) (u + 3) = 0; great--that makes perfect sense.1685

u times u is u2, plus 3u minus 4u--that is our -u; -4 times +3 is -12; it checks out.1694

So, we solve each one of these independently: u - 4 = 0, so we get u = 4.1701

Now, what is it?--because we are actually solving for the t; we are not solving for u.1706

So, we replace it with 4t = u; so we have 4t = 4, and there is only one thing that is going to end up giving us that.1711

It must be 41 there, so we have t = 1.1718

t = 1 is one of our answers; u + 3 = 0, so we have u = -3; we swap out for 4t...equals -3...1722

and at this point, we realize that that is madness; there is no possible way that 4 to the t can equal a negative number.1734

There is no real number that we can raise 4 to that is going to give us a negative number.1740

So, this way lies madness, so we knock it out; there are no possible answers there.1746

So, our only answer is t = 1; it seemed like we were going to get this answer, but then we realized that that is not possible.1750

If you want to check it out, we plug it in as a check: 42(1) - 41 = 12; 42 - 4 = 12; 16 - 4 = 12.1758

Indeed, that is true; so it checks out.1774

The final example: Solve for x exactly; then give an approximation.1777

We have the x that we are looking for; it is in the denominator.1782

We don't like things stuck in the denominator, so let's multiply it on both sides; we will multiply it by (ln(2x) - 3) on both sides.1785

So, we get 4 = 2 times (ln(2x) - 3) on both sides.1792

We can distribute the 2: 4 = 2 times (ln(2x) - 3); the thing we want to figure out is...it is like we are solving for ln(2x).1799

And then later, we will crack the log; but for now, let's just figure out solving for ln(2x), and then we can solve the log.1810

So, we see the -3 here; we add 3 to both sides, so we get 7 = 2 times...oops, I made a mistake!1817

An important thing to notice: it is 2 times (ln(2x) - 3), so it is 2 times -3; it is not minus 3; it is minus 6.1826

All right, so I have to always be careful with distribution, because it can catch anyone, including me and including your teachers.1836

You always have to watch out for distribution.1842

Add 6 to both sides; we get 10 = 2(ln(2x)); we divide by 2 on both sides; we get 5 = ln(2x).1844

We now raise both sides to the e; so e5, eln(2x).1855

Since these are the same base, natural log and e, they cancel out, and we have e5 = 2x.1861

So, we have that e5/2 = x; that is our exact answer, e5/2.1869

Now, e is this complicated irrational number; we can't really turn it into...it is not exactly a decimal number.1877

But sometimes you want to have a decimal approximation, because that makes it easier for us to work with things.1884

So, e5/2 is the exact answer; that is what it is precisely.1889

But if we want an approximate answer, e5/2 ends up being approximately 74.207; great.1893

Now, if we want to do a quick check, we can do a numerical approximation, where we try plugging it in using a calculator.1906

So, we would have 4 over the natural log of 2, times...replace our x, which we know is approximately 74.207, minus 3.1911

You work that through with a calculator, and you end up getting approximately 1.99999, and then it ends up changing after that.1923

But that is really, really close; so we know that this checks out numerically.1932

We have gotten that close to being exactly 2; so we see that numerically (because remember:1935

there is some rounding error when we get an approximation), with that little bit of rounding error, we are still extremely close.1940

So, we know that that is a good answer.1947

If we wanted to, we could also work it out and show that it ends up being a precise answer, as well--1949

that e5/2 is going to be equal to precisely what x has to be.1954

We can check by plugging that in: 4 over the natural log of 2, times e5 divided by 2, minus 3.1958

So, 2 times...dividing by 2...they cancel out; so we have 4 over the natural log of e5, minus 3.1970

e5 is going to end up being...4/ln(e5) has to be 5, because what do we raise e to1981

(since the natural log is base e) to get e5? Of course, we raise it to a 5 exponent.1991

5 - 3...we could also think about the fact that natural log is log base e, and we have an exponent base e, so they cancel each other out.1997

4 over 5 minus 3 becomes 2, which equals 2; that checks out.2006

So, we can do it either numerically or precisely, and see that it worked in either case.2011

All right, great; now I have a really good understanding of even probably the most complex kind2017

of logarithmic and exponential equations that will be thrown at you at this point in math.2021

So, with this sort of knowledge, you can go and do all kinds of problems, from the easy to the hard ones.2025

And you will be able to solve them if you follow these things.2030

Remember: be careful--mistakes happen; you even saw one happen to me.2032

So, they happen to everybody; it is really useful to do these checks.2036

By checking your work, you can make sure that you didn't make a mistake.2039

And if you see that something went wrong in the check, you can go through and carefully analyze your work and figure out where things went wrong.2042

All right, we will see you at Educator.com later--goodbye!2048