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INSTRUCTORS Carleen Eaton Grant Fraser
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Dr. Carleen Eaton

Dr. Carleen Eaton

Common Logarithms

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Table of Contents

I. Equations and Inequalities
Expressions and Formulas

22m 23s

Intro
0:00
Order of Operations
0:19
Variable
0:27
Algebraic Expression
0:46
Term
0:57
Example: Algebraic Expression
1:25
Evaluate Inside Grouping Symbols
1:55
Evaluate Powers
2:30
Multiply/Divide Left to Right
2:55
Add/Subtract Left to Right
3:35
Monomials
4:40
Examples of Monomials
4:52
Constant
5:27
Coefficient
5:46
Degree
6:25
Power
7:15
Polynomials
8:02
Examples of Polynomials
8:24
Binomials, Trinomials, Monomials
8:53
Term
9:21
Like Terms
10:02
Formulas
11:00
Example: Pythagorean Theorem
11:15
Example 1: Evaluate the Algebraic Expression
11:50
Example 2: Evaluate the Algebraic Expression
14:38
Example 3: Area of a Triangle
19:11
Example 4: Fahrenheit to Celsius
20:41
Properties of Real Numbers

20m 15s

Intro
0:00
Real Numbers
0:07
Number Line
0:15
Rational Numbers
0:46
Irrational Numbers
2:24
Venn Diagram of Real Numbers
4:03
Irrational Numbers
5:00
Rational Numbers
5:19
Real Number System
5:27
Natural Numbers
5:32
Whole Numbers
5:53
Integers
6:19
Fractions
6:46
Properties of Real Numbers
7:15
Commutative Property
7:34
Associative Property
8:07
Identity Property
9:04
Inverse Property
9:53
Distributive Property
11:03
Example 1: What Set of Numbers?
12:21
Example 2: What Properties Are Used?
13:56
Example 3: Multiplicative Inverse
16:00
Example 4: Simplify Using Properties
17:18
Solving Equations

19m 10s

Intro
0:00
Translations
0:06
Verbal Expressions and Algebraic Expressions
0:13
Example: Sum of Two Numbers
0:19
Example: Square of a Number
1:33
Properties of Equality
3:20
Reflexive Property
3:30
Symmetric Property
3:42
Transitive Property
4:01
Addition Property
5:01
Subtraction Property
5:37
Multiplication Property
6:02
Division Property
6:30
Solving Equations
6:58
Example: Using Properties
7:18
Solving for a Variable
8:25
Example: Solve for Z
8:34
Example 1: Write Algebraic Expression
10:15
Example 2: Write Verbal Expression
11:31
Example 3: Solve the Equation
14:05
Example 4: Simplify Using Properties
17:26
Solving Absolute Value Equations

17m 31s

Intro
0:00
Absolute Value Expressions
0:09
Distance from Zero
0:18
Example: Absolute Value Expression
0:24
Absolute Value Equations
1:50
Example: Absolute Value Equation
2:00
Example: Isolate Expression
3:13
No Solution
3:46
Empty Set
3:58
Example: No Solution
4:12
Number of Solutions
4:46
Check Each Solution
4:57
Example: Two Solutions
5:05
Example: No Solution
6:18
Example: One Solution
6:28
Example 1: Evaluate for X
7:16
Example 2: Write Verbal Expression
9:08
Example 3: Solve the Equation
12:18
Example 4: Simplify Using Properties
13:36
Solving Inequalities

17m 14s

Intro
0:00
Properties of Inequalities
0:08
Addition Property
0:17
Example: Using Numbers
0:30
Subtraction Property
1:03
Example: Using Numbers
1:19
Multiplication Properties
1:44
C>0 (Positive Number)
1:50
Example: Using Numbers
2:05
C<0 (Negative Number)
2:40
Example: Using Numbers
3:10
Division Properties
4:11
C>0 (Positive Number)
4:15
Example: Using Numbers
4:27
C<0 (Negative Number)
5:21
Example: Using Numbers
5:32
Describing the Solution Set
6:10
Example: Set Builder Notation
6:26
Example: Graph (Closed Circle)
7:08
Example: Graph (Open Circle)
7:30
Example 1: Solve the Inequality
7:58
Example 2: Solve the Inequality
9:06
Example 3: Solve the Inequality
10:10
Example 4: Solve the Inequality
13:12
Solving Compound and Absolute Value Inequalities

25m

Intro
0:00
Compound Inequalities
0:08
And and Or
0:13
Example: And
0:22
Example: Or
1:12
And Inequality
1:41
Intersection
1:49
Example: Numbers
2:08
Example: Inequality
2:43
Or Inequality
4:35
Example: Union
4:45
Example: Inequality
5:53
Absolute Value Inequalities
7:19
Definition of Absolute Value
7:33
Examples: Compound Inequalities
8:30
Example: Complex Inequality
12:21
Example 1: Solve the Inequality
12:54
Example 2: Solve the Inequality
17:21
Example 3: Solve the Inequality
18:54
Example 4: Solve the Inequality
22:15
II. Linear Relations and Functions
Relations and Functions

32m 5s

Intro
0:00
Coordinate Plane
0:20
X-Coordinate and Y-Coordinate
0:30
Example: Coordinate Pairs
0:37
Quadrants
1:20
Relations
2:14
Domain and Range
2:19
Set of Ordered Pairs
2:29
As a Table
2:51
Functions
4:21
One Element in Range
4:32
Example: Mapping
4:43
Example: Table and Map
6:26
One-to-One Functions
8:01
Example: One-to-One
8:22
Example: Not One-to-One
9:18
Graphs of Relations
11:01
Discrete and Continuous
11:12
Example: Discrete
11:22
Example: Continous
12:30
Vertical Line Test
14:09
Example: S Curve
14:29
Example: Function
16:15
Equations, Relations, and Functions
17:03
Independent Variable and Dependent Variable
17:16
Function Notation
19:11
Example: Function Notation
19:23
Example 1: Domain and Range
20:51
Example 2: Discrete or Continous
23:03
Example 3: Discrete or Continous
25:53
Example 4: Function Notation
30:05
Linear Equations

14m 46s

Intro
0:00
Linear Equations and Functions
0:07
Linear Equation
0:19
Example: Linear Equation
0:29
Example: Linear Function
1:07
Standard Form
2:02
Integer Constants with No Common Factor
2:08
Example: Standard Form
2:27
Graphing with Intercepts
4:05
X-Intercept and Y-Intercept
4:12
Example: Intercepts
4:26
Example: Graphing
5:14
Example 1: Linear Function
7:53
Example 2: Linear Function
9:10
Example 3: Standard Form
10:04
Example 4: Graph with Intercepts
12:25
Slope

23m 7s

Intro
0:00
Definition of Slope
0:07
Change in Y / Change in X
0:26
Example: Slope of Graph
0:37
Interpretation of Slope
3:07
Horizontal Line (0 Slope)
3:13
Vertical Line (Undefined Slope)
4:52
Rises to Right (Positive Slope)
6:36
Falls to Right (Negative Slope)
6:53
Parallel Lines
7:18
Example: Not Vertical
7:30
Example: Vertical
7:58
Perpendicular Lines
8:31
Example: Perpendicular
8:42
Example 1: Slope of Line
10:32
Example 2: Graph Line
11:45
Example 3: Parallel to Graph
13:37
Example 4: Perpendicular to Graph
17:57
Writing Linear Functions

23m 5s

Intro
0:00
Slope Intercept Form
0:11
m and b
0:28
Example: Graph Using Slope Intercept
0:43
Point Slope Form
2:41
Relation to Slope Formula
3:03
Example: Point Slope Form
4:36
Parallel and Perpendicular Lines
6:28
Review of Parallel and Perpendicular Lines
6:31
Example: Parallel
7:50
Example: Perpendicular
9:58
Example 1: Slope Intercept Form
11:07
Example 2: Slope Intercept Form
13:07
Example 3: Parallel
15:49
Example 4: Perpendicular
18:42
Special Functions

31m 5s

Intro
0:00
Step Functions
0:07
Example: Apple Prices
0:30
Absolute Value Function
4:55
Example: Absolute Value
5:05
Piecewise Functions
9:08
Example: Piecewise
9:27
Example 1: Absolute Value Function
14:00
Example 2: Absolute Value Function
20:39
Example 3: Piecewise Function
22:26
Example 4: Step Function
25:25
Graphing Inequalities

21m 42s

Intro
0:00
Graphing Linear Inequalities
0:07
Shaded Region
0:19
Using Test Points
0:32
Graph Corresponding Linear Function
0:46
Dashed or Solid Lines
0:59
Use Test Point
1:21
Example: Linear Inequality
1:58
Graphing Absolute Value Inequalities
4:50
Graph Corresponding Equations
4:59
Use Test Point
5:20
Example: Absolute Value Inequality
5:38
Example 1: Linear Inequality
9:17
Example 2: Linear Inequality
11:56
Example 3: Linear Inequality
14:29
Example 4: Absolute Value Inequality
17:06
III. Systems of Equations and Inequalities
Solving Systems of Equations by Graphing

17m 13s

Intro
0:00
Systems of Equations
0:09
Example: Two Equations
0:24
Solving by Graphing
0:53
Point of Intersection
1:09
Types of Systems
2:29
Independent (Single Solution)
2:34
Dependent (Infinite Solutions)
3:05
Inconsistent (No Solution)
4:23
Example 1: Solve by Graphing
5:20
Example 2: Solve by Graphing
9:10
Example 3: Solve by Graphing
12:27
Example 4: Solve by Graphing
14:54
Solving Systems of Equations Algebraically

23m 53s

Intro
0:00
Solving by Substitution
0:08
Example: System of Equations
0:36
Solving by Multiplication
7:22
Extra Step of Multiplying
7:38
Example: System of Equations
8:00
Inconsistent and Dependent Systems
11:14
Variables Drop Out
11:48
Inconsistent System (Never True)
12:01
Constant Equals Constant
12:53
Dependent System (Always True)
13:11
Example 1: Solve Algebraically
13:58
Example 2: Solve Algebraically
15:52
Example 3: Solve Algebraically
17:54
Example 4: Solve Algebraically
21:40
Solving Systems of Inequalities By Graphing

27m 12s

Intro
0:00
Solving by Graphing
0:08
Graph Each Inequality
0:25
Overlap
0:35
Corresponding Linear Equations
1:03
Test Point
1:23
Example: System of Inequalities
1:51
No Solution
7:06
Empty Set
7:26
Example: No Solution
7:34
Example 1: Solve by Graphing
10:27
Example 2: Solve by Graphing
13:30
Example 3: Solve by Graphing
17:19
Example 4: Solve by Graphing
23:23
Solving Systems of Equations in Three Variables

28m 53s

Intro
0:00
Solving Systems in Three Variables
0:17
Triple of Values
0:31
Example: Three Variables
0:56
Number of Solutions
5:55
One Solution
6:08
No Solution
6:24
Infinite Solutions
7:06
Example 1: Solve 3 Variables
7:59
Example 2: Solve 3 Variables
13:50
Example 3: Solve 3 Variables
19:54
Example 4: Solve 3 Variables
25:50
IV. Matrices
Basic Matrix Concepts

11m 34s

Intro
0:00
What is a Matrix
0:26
Brackets
0:46
Designation
1:21
Element
1:47
Matrix Equations
1:59
Dimensions
2:27
Rows (m) and Columns (n)
2:37
Examples: Dimensions
2:43
Special Matrices
4:22
Row Matrix
4:32
Column Matrix
5:00
Zero Matrix
6:00
Equal Matrices
6:30
Example: Corresponding Elements
6:36
Example 1: Matrix Dimension
8:12
Example 2: Matrix Dimension
9:03
Example 3: Zero Matrix
9:38
Example 4: Row and Column Matrix
10:26
Matrix Operations

21m 36s

Intro
0:00
Matrix Addition
0:18
Same Dimensions
0:25
Example: Adding Matrices
1:04
Matrix Subtraction
3:42
Same Dimensions
3:48
Example: Subtracting Matrices
4:04
Scalar Multiplication
6:08
Scalar Constant
6:24
Example: Multiplying Matrices
6:32
Properties of Matrix Operations
8:23
Commutative Property
8:41
Associative Property
9:08
Distributive Property
9:44
Example 1: Matrix Addition
10:24
Example 2: Matrix Subtraction
11:58
Example 3: Scalar Multiplication
14:23
Example 4: Matrix Properties
16:09
Matrix Multiplication

29m 36s

Intro
0:00
Dimension Requirement
0:17
n = p
0:24
Resulting Product Matrix (m x q)
1:21
Example: Multiplication
1:54
Matrix Multiplication
3:38
Example: Matrix Multiplication
4:07
Properties of Matrix Multiplication
10:46
Associative Property
11:00
Associative Property (Scalar)
11:28
Distributive Property
12:06
Distributive Property (Scalar)
12:30
Example 1: Possible Matrices
13:31
Example 2: Multiplying Matrices
17:08
Example 3: Multiplying Matrices
20:41
Example 4: Matrix Properties
24:41
Determinants

33m 13s

Intro
0:00
What is a Determinant
0:13
Square Matrices
0:23
Vertical Bars
0:41
Determinant of a 2x2 Matrix
1:21
Second Order Determinant
1:37
Formula
1:45
Example: 2x2 Determinant
1:58
Determinant of a 3x3 Matrix
2:50
Expansion by Minors
3:08
Third Order Determinant
3:19
Expanding Row One
4:06
Example: 3x3 Determinant
6:40
Diagonal Method for 3x3 Matrices
13:24
Example: Diagonal Method
13:36
Example 1: Determinant of 2x2
18:59
Example 2: Determinant of 3x3
20:03
Example 3: Determinant of 3x3
25:35
Example 4: Determinant of 3x3
29:22
Cramer's Rule

28m 25s

Intro
0:00
System of Two Equations in Two Variables
0:16
One Variable
0:50
Determinant of Denominator
1:14
Determinants of Numerators
2:23
Example: System of Equations
3:34
System of Three Equations in Three Variables
7:06
Determinant of Denominator
7:17
Determinants of Numerators
7:52
Example 1: Two Equations
8:57
Example 2: Two Equations
13:21
Example 3: Three Equations
17:11
Example 4: Three Equations
23:43
Identity and Inverse Matrices

22m 25s

Intro
0:00
Identity Matrix
0:13
Example: 2x2 Identity Matrix
0:30
Example: 4x4 Identity Matrix
0:50
Properties of Identity Matrices
1:24
Example: Multiplying Identity Matrix
2:52
Matrix Inverses
5:30
Writing Matrix Inverse
6:07
Inverse of a 2x2 Matrix
6:39
Example: 2x2 Matrix
7:31
Example 1: Inverse Matrix
10:18
Example 2: Find the Inverse Matrix
13:04
Example 3: Find the Inverse Matrix
17:53
Example 4: Find the Inverse Matrix
20:44
Solving Systems of Equations Using Matrices

22m 32s

Intro
0:00
Matrix Equations
0:11
Example: System of Equations
0:21
Solving Systems of Equations
4:01
Isolate x
4:16
Example: Using Numbers
5:10
Multiplicative Inverse
5:54
Example 1: Write as Matrix Equation
7:18
Example 2: Use Matrix Equations
9:12
Example 3: Use Matrix Equations
15:06
Example 4: Use Matrix Equations
19:35
V. Quadratic Functions and Inequalities
Graphing Quadratic Functions

31m 48s

Intro
0:00
Quadratic Functions
0:12
A is Zero
0:27
Example: Parabola
0:45
Properties of Parabolas
2:08
Axis of Symmetry
2:11
Vertex
2:32
Example: Parabola
2:48
Minimum and Maximum Values
9:02
Positive or Negative
9:28
Upward or Downward
9:58
Example: Minimum
10:31
Example: Maximum
11:16
Example 1: Axis of Symmetry, Vertex, Graph
12:41
Example 2: Axis of Symmetry, Vertex, Graph
17:25
Example 3: Minimum or Maximum
21:47
Example 4: Minimum or Maximum
27:09
Solving Quadratic Equations by Graphing

27m 3s

Intro
0:00
Quadratic Equations
0:16
Standard Form
0:18
Example: Quadratic Equation
0:47
Solving by Graphing
1:41
Roots (x-Intercepts)
1:48
Example: Number of Solutions
2:12
Estimating Solutions
9:23
Example: Integer Solutions
9:30
Example: Estimating
9:53
Example 1: Solve by Graphing
10:52
Example 2: Solve by Graphing
15:10
Example 1: Solve by Graphing
17:50
Example 1: Solve by Graphing
20:54
Solving Quadratic Equations by Factoring

19m 53s

Intro
0:00
Factoring Techniques
0:15
Greatest Common Factor (GCF)
0:37
Difference of Two Squares
1:48
Perfect Square Trinomials
2:30
General Trinomials
3:09
Zero Product Rule
5:22
Example: Zero Product
5:53
Example 1: Solve by Factoring
7:46
Example 1: Solve by Factoring
9:48
Example 1: Solve by Factoring
12:34
Example 1: Solve by Factoring
15:28
Imaginary and Complex Numbers

35m 45s

Intro
0:00
Properties of Square Roots
0:10
Product Property
0:26
Example: Product Property
0:56
Quotient Property
2:17
Example: Quotient Property
2:35
Imaginary Numbers
3:12
Imaginary i
3:51
Examples: Imaginary Number
4:22
Complex Numbers
7:23
Real Part and Imaginary Part
7:33
Examples: Complex Numbers
7:57
Equality
9:37
Example: Equal Complex Numbers
9:52
Addition and Subtraction
10:12
Examples: Adding Complex Numbers
10:25
Complex Plane
13:32
Horizontal Axis (Real)
13:49
Vertical Axis (Imaginary)
13:59
Example: Labeling
14:11
Multiplication
15:57
Example: FOIL Method
16:03
Division
18:37
Complex Conjugates
18:45
Conjugate Pairs
19:10
Example: Dividing Complex Numbers
20:00
Example 1: Simplify Complex Number
24:50
Example 2: Simplify Complex Number
27:56
Example 3: Multiply Complex Numbers
29:27
Example 3: Dividing Complex Numbers
31:48
Completing the Square

27m 11s

Intro
0:00
Square Root Property
0:12
Example: Perfect Square
0:38
Example: Perfect Square Trinomial
3:00
Completing the Square
4:39
Constant Term
4:50
Example: Complete the Square
5:04
Solve Equations
6:42
Add to Both Sides
6:59
Example: Complete the Square
7:07
Equations Where a Not Equal to 1
10:58
Divide by Coefficient
11:08
Example: Complete the Square
11:24
Complex Solutions
14:05
Real and Imaginary
14:14
Example: Complex Solution
14:35
Example 1: Square Root Property
18:31
Example 2: Complete the Square
19:15
Example 3: Complete the Square
20:40
Example 4: Complete the Square
23:56
Quadratic Formula and the Discriminant

22m 48s

Intro
0:00
Quadratic Formula
0:21
Standard Form
0:29
Example: Quadratic Formula
0:57
One Rational Root
3:00
Example: One Root
3:31
Complex Solutions
6:16
Complex Conjugate
6:28
Example: Complex Solution
7:15
Discriminant
9:42
Positive Discriminant
10:03
Perfect Square (Rational)
10:51
Not Perfect Square (2 Irrational)
11:27
Negative Discriminant
12:28
Zero Discriminant
12:57
Example 1: Quadratic Formula
13:50
Example 2: Quadratic Formula
16:03
Example 3: Quadratic Formula
19:00
Example 4: Discriminant
21:33
Analyzing the Graphs of Quadratic Functions

30m 7s

Intro
0:00
Vertex Form
0:12
H and K
0:32
Axis of Symmetry
0:36
Vertex
0:42
Example: Origin
1:00
Example: k = 2
2:12
Example: h = 1
4:27
Significance of Coefficient a
7:13
Example: |a| > 1
7:25
Example: |a| < 1
8:18
Example: |a| > 0
8:51
Example: |a| < 0
9:05
Writing Quadratic Equations in Vertex Form
10:22
Standard Form to Vertex Form
10:35
Example: Standard Form
11:02
Example: a Term Not 1
14:42
Example 1: Vertex Form
19:47
Example 2: Vertex Form
22:09
Example 3: Vertex Form
24:32
Example 4: Vertex Form
28:23
Graphing and Solving Quadratic Inequalities

27m 5s

Intro
0:00
Graphing Quadratic Inequalities
0:11
Test Point
0:18
Example: Quadratic Inequality
0:29
Solving Quadratic Inequalities
3:57
Example: Parameter
4:24
Example 1: Graph Inequality
11:16
Example 2: Solve Inequality
14:27
Example 3: Graph Inequality
19:14
Example 4: Solve Inequality
23:48
VI. Polynomial Functions
Properties of Exponents

19m 29s

Intro
0:00
Simplifying Exponential Expressions
0:09
Monomial Simplest Form
0:19
Negative Exponents
1:07
Examples: Simple
1:34
Properties of Exponents
3:06
Negative Exponents
3:13
Mutliplying Same Base
3:24
Dividing Same Base
3:45
Raising Power to a Power
4:33
Parentheses (Multiplying)
5:11
Parentheses (Dividing)
5:47
Raising to 0th Power
6:15
Example 1: Simplify Exponents
7:59
Example 2: Simplify Exponents
10:41
Example 3: Simplify Exponents
14:11
Example 4: Simplify Exponents
18:04
Operations on Polynomials

13m 27s

Intro
0:00
Adding and Subtracting Polynomials
0:13
Like Terms and Like Monomials
0:23
Examples: Adding Monomials
1:14
Multiplying Polynomials
3:40
Distributive Property
3:44
Example: Monomial by Polynomial
4:06
Example 1: Simplify Polynomials
5:47
Example 2: Simplify Polynomials
6:28
Example 3: Simplify Polynomials
8:38
Example 4: Simplify Polynomials
10:47
Dividing Polynomials

31m 11s

Intro
0:00
Dividing by a Monomial
0:13
Example: Numbers
0:26
Example: Polynomial by a Monomial
1:18
Long Division
2:28
Remainder Term
2:41
Example: Dividing with Numbers
3:04
Example: With Polynomials
5:01
Example: Missing Terms
7:58
Synthetic Division
11:44
Restriction
12:04
Example: Divisor in Form
12:20
Divisor in Synthetic Division
15:54
Example: Coefficient to 1
16:07
Example 1: Divide Polynomials
17:10
Example 2: Divide Polynomials
19:08
Example 3: Synthetic Division
21:42
Example 4: Synthetic Division
25:09
Polynomial Functions

22m 30s

Intro
0:00
Polynomial in One Variable
0:13
Leading Coefficient
0:27
Example: Polynomial
1:18
Degree
1:31
Polynomial Functions
2:57
Example: Function
3:13
Function Values
3:33
Example: Numerical Values
3:53
Example: Algebraic Expressions
5:11
Zeros of Polynomial Functions
5:50
Odd Degree
6:04
Even Degree
7:29
End Behavior
8:28
Even Degrees
9:09
Example: Leading Coefficient +/-
9:23
Odd Degrees
12:51
Example: Leading Coefficient +/-
13:00
Example 1: Degree and Leading Coefficient
15:03
Example 2: Polynomial Function
15:56
Example 3: Polynomial Function
17:34
Example 4: End Behavior
19:53
Analyzing Graphs of Polynomial Functions

33m 29s

Intro
0:00
Graphing Polynomial Functions
0:11
Example: Table and End Behavior
0:39
Location Principle
4:43
Zero Between Two Points
5:03
Example: Location Principle
5:21
Maximum and Minimum Points
8:40
Relative Maximum and Relative Minimum
9:16
Example: Number of Relative Max/Min
11:11
Example 1: Graph Polynomial Function
11:57
Example 2: Graph Polynomial Function
16:19
Example 3: Graph Polynomial Function
23:27
Example 4: Graph Polynomial Function
28:35
Solving Polynomial Functions

21m 10s

Intro
0:00
Factoring Polynomials
0:06
Greatest Common Factor (GCF)
0:25
Difference of Two Squares
1:14
Perfect Square Trinomials
2:07
General Trinomials
2:57
Grouping
4:32
Sum and Difference of Two Cubes
6:03
Examples: Two Cubes
6:14
Quadratic Form
8:22
Example: Quadratic Form
8:44
Example 1: Factor Polynomial
12:03
Example 2: Factor Polynomial
13:54
Example 3: Quadratic Form
15:33
Example 4: Solve Polynomial Function
17:24
Remainder and Factor Theorems

31m 21s

Intro
0:00
Remainder Theorem
0:07
Checking Work
0:22
Dividend and Divisor in Theorem
1:12
Example: f(a)
2:05
Synthetic Substitution
5:43
Example: Polynomial Function
6:15
Factor Theorem
9:54
Example: Numbers
10:16
Example: Confirm Factor
11:27
Factoring Polynomials
14:48
Example: 3rd Degree Polynomial
15:07
Example 1: Remainder Theorem
19:17
Example 2: Other Factors
21:57
Example 3: Remainder Theorem
25:52
Example 4: Other Factors
28:21
Roots and Zeros

31m 27s

Intro
0:00
Number of Roots
0:08
Not Nature of Roots
0:18
Example: Real and Complex Roots
0:25
Descartes' Rule of Signs
2:05
Positive Real Roots
2:21
Example: Positve
2:39
Negative Real Roots
5:44
Example: Negative
6:06
Finding the Roots
9:59
Example: Combination of Real and Complex
10:07
Conjugate Roots
13:18
Example: Conjugate Roots
13:50
Example 1: Solve Polynomial
16:03
Example 2: Solve Polynomial
18:36
Example 3: Possible Combinations
23:13
Example 4: Possible Combinations
27:11
Rational Zero Theorem

31m 16s

Intro
0:00
Equation
0:08
List of Possibilities
0:16
Equation with Constant and Leading Coefficient
1:04
Example: Rational Zero
2:46
Leading Coefficient Equal to One
7:19
Equation with Leading Coefficient of One
7:34
Example: Coefficient Equal to 1
8:45
Finding Rational Zeros
12:58
Division with Remainder Zero
13:32
Example 1: Possible Rational Zeros
14:20
Example 2: Possible Rational Zeros
16:02
Example 3: Possible Rational Zeros
19:58
Example 4: Find All Zeros
22:06
VII. Radical Expressions and Inequalities
Operations on Functions

34m 30s

Intro
0:00
Arithmetic Operations
0:07
Domain
0:16
Intersection
0:24
Denominator is Zero
0:49
Example: Operations
1:02
Composition of Functions
7:18
Notation
7:48
Right to Left
8:18
Example: Composition
8:48
Composition is Not Commutative
17:23
Example: Not Commutative
17:51
Example 1: Function Operations
20:55
Example 2: Function Operations
24:34
Example 3: Compositions
27:51
Example 4: Function Operations
31:09
Inverse Functions and Relations

22m 42s

Intro
0:00
Inverse of a Relation
0:14
Example: Ordered Pairs
0:56
Inverse of a Function
3:24
Domain and Range Switched
3:52
Example: Inverse
4:28
Procedure to Construct an Inverse Function
6:42
f(x) to y
6:42
Interchange x and y
6:59
Solve for y
7:06
Write Inverse f(x) for y
7:14
Example: Inverse Function
7:25
Example: Inverse Function 2
8:48
Inverses and Compositions
10:44
Example: Inverse Composition
11:46
Example 1: Inverse Relation
14:49
Example 2: Inverse of Function
15:40
Example 3: Inverse of Function
17:06
Example 4: Inverse Functions
18:55
Square Root Functions and Inequalities

30m 4s

Intro
0:00
Square Root Functions
0:07
Examples: Square Root Function
0:16
Example: Not Square Root Function
0:46
Radicand
1:12
Example: Restriction
1:31
Graphing Square Root Functions
3:42
Example: Graphing
3:49
Square Root Inequalities
8:47
Same Technique
9:00
Example: Square Root Inequality
9:20
Example 1: Graph Square Root Function
15:19
Example 2: Graph Square Root Function
18:03
Example 3: Graph Square Root Function
22:41
Example 4: Square Root Inequalities
25:37
nth Roots

20m 46s

Intro
0:00
Definition of the nth Root
0:07
Example: 5th Root
0:20
Example: 6th Root
0:51
Principal nth Root
1:39
Example: Principal Roots
2:06
Using Absolute Values
5:58
Example: Square Root
6:18
Example: 6th Root
8:40
Example: Negative
10:15
Example 1: Simplify Radicals
12:23
Example 2: Simplify Radicals
13:29
Example 3: Simplify Radicals
16:07
Example 4: Simplify Radicals
18:18
Operations with Radical Expressions

41m 11s

Intro
0:00
Properties of Radicals
0:16
Quotient Property
0:29
Example: Quotient
1:00
Example: Product Property
1:47
Simplifying Radical Expressions
3:24
Radicand No nth Powers
3:47
Radicand No Fractions
6:33
No Radicals in Denominator
7:16
Rationalizing Denominators
8:27
Example: Radicand nth Power
9:05
Conjugate Radical Expressions
11:47
Conjugates
12:07
Example: Conjugate Radical Expression
13:11
Adding and Subtracting Radicals
16:12
Same Index, Same Radicand
16:20
Example: Like Radicals
16:28
Multiplying Radicals
19:04
Distributive Property
19:10
Example: Multiplying Radicals
19:20
Example 1: Simplify Radical
24:11
Example 2: Simplify Radicals
28:43
Example 3: Simplify Radicals
32:00
Example 4: Simplify Radical
36:34
Rational Exponents

30m 45s

Intro
0:00
Definition 1
0:20
Example: Using Numbers
0:39
Example: Non-Negative
2:46
Example: Odd
3:34
Definition 2
4:32
Restriction
4:52
Example: Relate to Definition 1
5:04
Example: m Not 1
5:31
Simplifying Expressions
7:53
Multiplication
8:31
Division
9:29
Multiply Exponents
10:08
Raised Power
11:05
Zero Power
11:29
Negative Power
11:49
Simplified Form
13:52
Complex Fraction
14:16
Negative Exponents
14:40
Example: More Complicated
15:14
Example 1: Write as Radical
19:03
Example 2: Write with Rational Exponents
20:40
Example 3: Complex Fraction
22:09
Example 4: Complex Fraction
26:22
Solving Radical Equations and Inequalities

31m 27s

Intro
0:00
Radical Equations
0:11
Variables in Radicands
0:22
Example: Radical Equation
1:06
Example: Complex Equation
2:42
Extraneous Roots
7:21
Squaring Technique
7:35
Double Check
7:44
Example: Extraneous
8:21
Eliminating nth Roots
10:04
Isolate and Raise Power
10:14
Example: nth Root
10:27
Radical Inequalities
11:27
Restriction: Index is Even
11:53
Example: Radical Inequality
12:29
Example 1: Solve Radical Equation
15:41
Example 2: Solve Radical Equation
17:44
Example 3: Solve Radical Inequality
20:24
Example 4: Solve Radical Equation
24:34
VIII. Rational Equations and Inequalities
Multiplying and Dividing Rational Expressions

40m 54s

Intro
0:00
Simplifying Rational Expressions
0:22
Algebraic Fraction
0:29
Examples: Rational Expressions
0:49
Example: GCF
1:33
Example: Simplify Rational Expression
2:26
Factoring -1
4:04
Example: Simplify with -1
4:19
Multiplying and Dividing Rational Expressions
6:59
Multiplying and Dividing
7:28
Example: Multiplying Rational Expressions
8:36
Example: Dividing Rational Expressions
11:20
Factoring
14:01
Factoring Polynomials
14:19
Example: Factoring
14:35
Complex Fractions
18:22
Example: Numbers
18:37
Example: Algebraic Complex Fractions
19:25
Example 1: Simplify Rational Expression
25:56
Example 2: Simplify Rational Expression
29:34
Example 3: Simplify Rational Expression
31:39
Example 4: Simplify Rational Expression
37:50
Adding and Subtracting Rational Expressions

55m 4s

Intro
0:00
Least Common Multiple (LCM)
0:27
Examples: LCM of Numbers
0:43
Example: LCM of Polynomials
4:02
Adding and Subtracting
7:55
Least Common Denominator (LCD)
8:07
Example: Numbers
8:17
Example: Rational Expressions
11:03
Equivalent Fractions
15:22
Simplifying Complex Fractions
21:19
Example: Previous Lessons
21:36
Example: More Complex
22:53
Example 1: Find LCM
28:30
Example 2: Add Rational Expressions
31:44
Example 3: Subtract Rational Expressions
39:18
Example 4: Simplify Rational Expression
38:26
Graphing Rational Functions

57m 13s

Intro
0:00
Rational Functions
0:18
Restriction
0:34
Example: Rational Function
0:51
Breaks in Continuity
2:52
Example: Continuous Function
3:10
Discontinuities
3:30
Example: Excluded Values
4:37
Graphs and Discontinuities
5:02
Common Binomial Factor (Hole)
5:08
Example: Common Factor
5:31
Asymptote
10:06
Example: Vertical Asymptote
11:08
Horizontal Asymptotes
20:00
Example: Horizontal Asymptote
20:25
Example 1: Holes and Vertical Asymptotes
26:12
Example 2: Graph Rational Faction
28:35
Example 3: Graph Rational Faction
39:23
Example 4: Graph Rational Faction
47:28
Direct, Joint, and Inverse Variation

20m 21s

Intro
0:00
Direct Variation
0:07
Constant of Variation
0:25
Graph of Constant Variation
1:26
Slope is Constant k
1:35
Example: Straight Lines
1:41
Joint Variation
2:48
Three Variables
2:52
Inverse Variation
3:38
Rewritten Form
3:52
Examples in Biology
4:22
Graph of Inverse Variation
4:51
Asymptotes are Axes
5:12
Example: Inverse Variation
5:40
Proportions
10:11
Direct Variation
10:25
Inverse Variation
11:32
Example 1: Type of Variation
12:42
Example 2: Direct Variation
14:13
Example 3: Joint Variation
16:24
Example 4: Graph Rational Faction
18:50
Solving Rational Equations and Inequalities

55m 14s

Intro
0:00
Rational Equations
0:15
Example: Algebraic Fraction
0:26
Least Common Denominator
0:49
Example: Simple Rational Equation
1:22
Example: Solve Rational Equation
5:40
Extraneous Solutions
9:31
Doublecheck
10:00
No Solution
10:38
Example: Extraneous
10:44
Rational Inequalities
14:01
Excluded Values
14:31
Solve Related Equation
14:49
Find Intervals
14:58
Use Test Values
15:25
Example: Rational Inequality
15:51
Example: Rational Inequality 2
17:07
Example 1: Rational Equation
28:50
Example 2: Rational Equation
33:51
Example 3: Rational Equation
38:19
Example 4: Rational Inequality
46:49
IX. Exponential and Logarithmic Relations
Exponential Functions

35m 58s

Intro
0:00
What is an Exponential Function?
0:12
Restriction on b
0:31
Base
0:46
Example: Exponents as Bases
0:56
Variables as Exponents
1:12
Example: Exponential Function
1:50
Graphing Exponential Functions
2:33
Example: Using Table
2:49
Properties
11:52
Continuous and One to One
12:00
Domain is All Real Numbers
13:14
X-Axis Asymptote
13:55
Y-Intercept
14:02
Reflection Across Y-Axis
14:31
Growth and Decay
15:06
Exponential Growth
15:10
Real Life Examples
15:41
Example: Growth
15:52
Example: Decay
16:12
Real Life Examples
16:30
Equations
17:32
Bases are Same
18:05
Examples: Variables as Exponents
18:20
Inequalities
21:29
Property
21:51
Example: Inequality
22:37
Example 1: Graph Exponential Function
24:05
Example 2: Growth or Decay
27:50
Example 3: Exponential Equation
29:31
Example 4: Exponential Inequality
32:54
Logarithms and Logarithmic Functions

45m 54s

Intro
0:00
What are Logarithms?
0:08
Restrictions
0:15
Written Form
0:26
Logarithms are Exponents
0:52
Example: Logarithms
1:49
Logarithmic Functions
5:14
Same Restrictions
5:30
Inverses
5:53
Example: Logarithmic Function
6:24
Graph of the Logarithmic Function
9:20
Example: Using Table
9:35
Properties
15:09
Continuous and One to One
15:14
Domain
15:36
Range
15:56
Y-Axis is Asymptote
16:02
X Intercept
16:12
Inverse Property
16:57
Compositions of Functions
17:10
Equations
18:30
Example: Logarithmic Equation
19:13
Inequalities
20:36
Properties
20:47
Example: Logarithmic Inequality
21:40
Equations with Logarithms on Both Sides
24:43
Property
24:51
Example: Both Sides
25:23
Inequalities with Logarithms on Both Sides
26:52
Property
27:02
Example: Both Sides
28:05
Example 1: Solve Log Equation
31:52
Example 2: Solve Log Equation
33:53
Example 3: Solve Log Equation
36:15
Example 4: Solve Log Inequality
39:19
Properties of Logarithms

28m 43s

Intro
0:00
Product Property
0:08
Example: Product
0:46
Quotient Property
2:40
Example: Quotient
2:59
Power Property
3:51
Moved Exponent
4:07
Example: Power
4:37
Equations
5:15
Example: Use Properties
5:58
Example 1: Simplify Log
11:17
Example 2: Single Log
15:54
Example 3: Solve Log Equation
18:48
Example 4: Solve Log Equation
22:13
Common Logarithms

25m 23s

Intro
0:00
What are Common Logarithms?
0:10
Real World Applications
0:16
Base Not Written
0:27
Example: Base 10
0:39
Equations
1:47
Example: Same Base
1:56
Example: Different Base
2:37
Inequalities
6:07
Multiplying/Dividing Inequality
6:21
Example: Log Inequality
6:54
Change of Base
12:45
Base 10
13:24
Example: Change of Base
14:05
Example 1: Log Equation
15:21
Example 2: Common Logs
17:13
Example 3: Log Equation
18:22
Example 4: Log Inequality
21:52
Base e and Natural Logarithms

21m 14s

Intro
0:00
Number e
0:09
Natural Base
0:21
Growth/Decay
0:33
Example: Exponential Function
0:53
Natural Logarithms
1:11
ln x
1:19
Inverse and Identity Function
1:39
Example: Inverse Composition
1:55
Equations and Inequalities
4:39
Extraneous Solutions
5:30
Examples: Natural Log Equations
5:48
Example 1: Natural Log Equation
9:08
Example 2: Natural Log Equation
10:37
Example 3: Natural Log Inequality
16:54
Example 4: Natural Log Inequality
18:16
Exponential Growth and Decay

24m 30s

Intro
0:00
Decay
0:17
Decreases by Fixed Percentage
0:23
Rate of Decay
0:56
Example: Finance
1:34
Scientific Model of Decay
3:37
Exponential Decay
3:45
Radioactive Decay
4:13
Example: Half Life
5:33
Growth
9:06
Increases by Fixed Percentage
9:18
Example: Finance
10:09
Scientific Model of Growth
11:35
Population Growth
12:04
Example: Growth
12:20
Example 1: Computer Price
14:00
Example 2: Stock Price
15:46
Example 3: Medicine Disintegration
19:10
Example 4: Population Growth
22:33
X. Conic Sections
Midpoint and Distance Formulas

32m 42s

Intro
0:00
Midpoint Formula
0:15
Example: Midpoint
0:30
Distance Formula
2:30
Example: Distance
2:52
Example 1: Midpoint and Distance
4:58
Example 2: Midpoint and Distance
8:07
Example 3: Median Length
18:51
Example 4: Perimeter and Area
23:36
Parabolas

41m 27s

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

21m 3s

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

46m 51s

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

38m 15s

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

18m 43s

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

47m 4s

Intro
0:00
Linear Quadratic Systems
0:22
Example: Linear Quadratic System
0:45
Solutions
2:49
Graphs of Possible Solutions
3:10
Quadratic Quadratic System
4:10
Example: Elimination
4:21
Solutions
11:39
Example: 0, 1, 2, 3, 4 Solutions
11:50
Systems of Quadratic Inequalities
12:48
Example: Quadratic Inequality
13:09
Example 1: Solve Quadratic System
21:42
Example 2: Solve Quadratic System
29:13
Example 3: Solve Quadratic System
35:02
Example 4: Solve Quadratic Inequality
40:29
XI. Sequences and Series
Arithmetic Sequences

21m 16s

Intro
0:00
Sequences
0:10
General Form of Sequence
0:16
Example: Finite/Infinite Sequences
0:33
Arithmetic Sequences
0:28
Common Difference
2:41
Example: Arithmetic Sequence
2:50
Formula for the nth Term
3:51
Example: nth Term
4:32
Equation for the nth Term
6:37
Example: Using Formula
6:56
Arithmetic Means
9:47
Example: Arithmetic Means
10:16
Example 1: nth Term
12:38
Example 2: Arithmetic Means
13:49
Example 3: Arithmetic Means
16:12
Example 4: nth Term
18:26
Arithmetic Series

21m 36s

Intro
0:00
What are Arithmetic Series?
0:11
Common Difference
0:28
Example: Arithmetic Sequence
0:43
Example: Arithmetic Series
1:09
Finite/Infinite Series
1:36
Sum of Arithmetic Series
2:27
Example: Sum
3:21
Sigma Notation
5:53
Index
6:14
Example: Sigma Notation
7:14
Example 1: First Term
9:00
Example 2: Three Terms
10:52
Example 3: Sum of Series
14:14
Example 4: Sum of Series
18:13
Geometric Sequences

23m 3s

Intro
0:00
Geometric Sequences
0:11
Common Difference
0:38
Common Ratio
1:08
Example: Geometric Sequence
2:38
nth Term of a Geometric Sequence
4:41
Example: nth Term
4:56
Geometric Means
6:51
Example: Geometric Mean
7:09
Example 1: 9th Term
12:04
Example 2: Geometric Means
15:18
Example 3: nth Term
18:32
Example 4: Three Terms
20:59
Geometric Series

22m 43s

Intro
0:00
What are Geometric Series?
0:11
List of Numbers
0:24
Example: Geometric Series
1:12
Sum of Geometric Series
2:16
Example: Sum of Geometric Series
2:41
Sigma Notation
4:21
Lower Index, Upper Index
4:38
Example: Sigma Notation
4:57
Another Sum Formula
6:08
Example: n Unknown
6:28
Specific Terms
7:41
Sum Formula
7:56
Example: Specific Term
8:11
Example 1: Sum of Geometric Series
10:02
Example 2: Sum of 8 Terms
14:15
Example 3: Sum of Geometric Series
18:23
Example 4: First Term
20:16
Infinite Geometric Series

18m 32s

Intro
0:00
What are Infinite Geometric Series
0:10
Example: Finite
0:29
Example: Infinite
0:51
Partial Sums
1:09
Formula
1:37
Sum of an Infinite Geometric Series
2:39
Convergent Series
2:58
Example: Sum of Convergent Series
3:28
Sigma Notation
7:31
Example: Sigma
8:17
Repeating Decimals
8:42
Example: Repeating Decimal
8:53
Example 1: Sum of Infinite Geometric Series
12:15
Example 2: Repeating Decimal
13:24
Example 3: Sum of Infinite Geometric Series
15:14
Example 4: Repeating Decimal
16:48
Recursion and Special Sequences

14m 34s

Intro
0:00
Fibonacci Sequence
0:05
Background of Fibonacci
0:23
Recursive Formula
0:37
Fibonacci Sequence
0:52
Example: Recursive Formula
2:18
Iteration
3:49
Example: Iteration
4:30
Example 1: Five Terms
7:08
Example 2: Three Terms
9:00
Example 3: Five Terms
10:38
Example 4: Three Iterates
12:41
Binomial Theorem

48m 30s

Intro
0:00
Pascal's Triangle
0:06
Expand Binomial
0:13
Pascal's Triangle
4:26
Properties
6:52
Example: Properties of Binomials
6:58
Factorials
9:11
Product
9:28
Example: Factorial
9:45
Binomial Theorem
11:08
Example: Binomial Theorem
13:48
Finding a Specific Term
18:36
Example: Specific Term
19:26
Example 1: Expand
24:39
Example 2: Fourth Term
30:26
Example 3: Five Terms
36:13
Example 4: Three Iterates
45:07
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Lecture Comments (18)

1 answer

Last reply by: Dr Carleen Eaton
Sat Nov 7, 2015 6:12 PM

Post by Peter Ke on October 24, 2015

For Example 2, shouldn't the answer be:

Log(base10)22 / Log(base10)7 ?

1 answer

Last reply by: Dr Carleen Eaton
Fri Nov 22, 2013 7:39 PM

Post by Juan Herrera on November 21, 2013

At 15:05, Log base 6 of 8 want to write as Log base 10
** It should be: Log base 10 of 8 divided by Log base 10 of 6

I believe the main purpose of this property is to change
difficult Logarithmic bases to a common (base 10) or Natural base (e)
to solve problems in a practical easy way.

1 answer

Last reply by: Dr Carleen Eaton
Fri Mar 1, 2013 10:59 PM

Post by Kenneth Montfort on February 27, 2013

Do the log properties not apply here to common logs. For example, if you have log 7/log 4, can't I just subtract 7 and 4 to find the final log value (e.g. log 3)? Or is it acceptable to just leave it in the form log 7/log 4.

1 answer

Last reply by: Dr Carleen Eaton
Sat Aug 25, 2012 12:34 PM

Post by Jorge Sardinas on August 14, 2012

in example one at 2x=log19/log3+3 your supposed to divide two by everything

1 answer

Last reply by: Dr Carleen Eaton
Thu Jun 21, 2012 9:56 PM

Post by Rob Escalera on June 19, 2012

It seems, that as long as you take the entire quantity, you can either divide or distribute and the answer will be the same.

1 answer

Last reply by: Dr Carleen Eaton
Thu Jun 21, 2012 9:55 PM

Post by Rob Escalera on June 18, 2012

In example I again, don't you need to distribute the quantity (2x-3) log3 ?

1 answer

Last reply by: Dr Carleen Eaton
Thu Jun 21, 2012 9:50 PM

Post by Rob Escalera on June 18, 2012

At 4:00, don't you need to distribute the (3x 4)log2 to become: 3xlog2 4log2 = log5?

Which should then be: 3xlog2 = log5 - 4log2 =>
x(3log2) = log5 - 4log2 => x = (log5 - 4log2)/3log2

3 answers

Last reply by: Dr Carleen Eaton
Sun Mar 11, 2012 7:21 PM

Post by Jimmi Aastrom on March 29, 2011

A short remark.
At around 5:30 I believe that a paranthases needs to be added arount the log-fraction and negative 4!?

1/3(Log5/Log2)-4 needs to be 1/3((Log5/Log2)-4)

Otherwise only part of the expression is devided by 3 which would be incorrect.

Common Logarithms

  • Solve exponential equations by taking the common log of both sides of the equation. You can then use a calculator to get a decimal approximation of the answer.
  • If you want to get a decimal approximation of a logarithmic expression, convert the log expression to a log expression to the base 10 using the change of base formula. Then use a calculator to evaluate the new expression.

Common Logarithms

Solve 109x − 6 = 2
  • Step 1) Take the common log of both sides
  • 109x − 6 = 2 = > log(109x − 6) = log(2)
  • Step 2) Bring the exponents down using properties of logs
  • log(109x − 6) = log(2)
  • (9x − 6)log(10) = log(2)
  • Step3: Distribute using distributive property
  • 9x*log(10) − 6(log(10) = log(2)
  • Step 4 - Add 6log(10) to both sides
  • 9xlog(10) = log(2) + 6log(10)
  • Step 4 - Divide both sides by 9log(10)
  • x = [(log(2) + 6log(10))/9log(10)]
  • Use a calculator to evaluate
x = 0.7001
Solve 10 − 6x − 3 = 84
  • Step 1) Take the common log of both sides
  • 10 − 6x − 3 = 84 = > log(10 − 6x − 3) = log(84)
  • Step 2) Bring the exponents down using properties of logs
  • log(10 − 6x − 3) = log(84)
  • ( − 6x − 3)log(10) = log(84)
  • Step3: Distribute using distributive property
  • − 6x*log(10) − 3(log(10) = log(84)
  • Step 4 - Add 3log(10) to both sides
  • − 6xlog(10) = log(84) + 3log(10)
  • Step 4 - Divide both sides by − 6log(10)
  • x = [(log(84) + 3log(10))/( − 6log(10))]
  • Use a calculator to evaluate
x = − 0.8207
Solve 9*38 − 6x − 7 = 41
  • Step 1) Add seven to both sides
  • 9*38 − 6x = 48
  • Step 2) Divide both sides by nine
  • 38 − 6x = [48/9]
  • Step 3) Take the common log of both sides
  • log(38 − 6x) = log[48/9]
  • Step 4) Bring the exponents down using properties of logs
  • log(38 − 6x) = log[48/9]
  • (8 − 6x)log(3) = log([48/9])
  • Step 5) Distribute using distributive property
  • 8*log(3) − 6x(log(3)) = log([48/9])
  • Step 6) - Subtract 8log(3) from both sides
  • − 6x(log(3)) = log([48/9]) − 8*log(3)
  • Step 7) Divide both sides by − 6log(2)
  • x = [(log([48/9]) − 8*log(3))/( − 6log(3))]
  • Use a calculator to evaluate
x = 1.0794
Solve − 3*186 − 8x + 10 = − 10
  • Step 1) Subtract 10 from both sides
  • − 3*186 − 8x = − 20
  • Step 2) Divide both sides by − 3
  • 186 − 8x = [20/3]
  • Step 3) Take the common log of both sides
  • log(186 − 8x) = log[20/3]
  • Step 4) Bring the exponents down using properties of logs
  • log(186 − 8x) = log[20/3]
  • (6 − 8x)log(18) = log([20/3])
  • Step 5) Distribute using distributive property
  • 6log(18) − 8xlog(18) = log([20/3])
  • Step 6) - Subtract 6log(3) from both sides
  • − 8xlog(18) = log([20/3]) − 6log(18)
  • Step 7) Divide both sides by − 8log(18)
  • x = [(log([20/3]) − 6log(18))/( − 8log(18))]
  • Use a calculator to evaluate
x = 0.6680
Solve − 4*156x + 4 + 9 = − 65
  • Step 1) Subtract 9 from both sides
  • − 4*156x + 4 = − 74
  • Step 2) Divide both sides by − 4
  • 156x + 4 = [74/4]
  • 156x + 4 = [37/2]
  • Step 3) Take the common log of both sides
  • log(156x + 4) = log[37/2]
  • Step 4) Bring the exponents down using properties of logs
  • (6x + 4)log(15) = log[37/2]
  • Step 5) Distribute using distributive property
  • 6xlog(15) + 4log(15) = log[37/2]
  • Step 6) - Subtract 4log(15) from both sides
  • 6xlog(15) = log[37/2] − 4log(15)
  • Step 7) Divide both sides by 6log(15)
  • x = [(log[37/2] − 4log(15))/6log(15)]
  • Use a calculator to evaluate
x = − 0.4871
Solve 9*66 − 10x + 1 = 67
  • Step 1) Subtract 1 from both sides
  • 9*66 − 10x = 66
  • Step 2) Divide both sides by 9
  • 66 − 10x = [66/9]
  • 66 − 10x = [22/3]
  • Step 3) Take the common log of both sides
  • log(66 − 10x) = log[22/3]
  • Step 4) Bring the exponents down using properties of logs
  • (6 − 10x)log(6) = log[22/3]
  • Step 5) Distribute using distributive property
  • 6log(6) − 10xlog(6) = log[22/3]
  • Step 6) - Subtract 6log(6) from both sides
  • − 10xlog(6) = log[22/3] − 6log(6)
  • Step 7) Divide both sides by − 10log(6)
  • x = [(log[22/3] − 6log(6))/( − 10log(6))]
  • Use a calculator to evaluate
x = 0.4888
Write using common logarithms:
log327
  • We're going to use the change of base formula which states that
  • logax = [(logbx)/(logba)]
  • log1027 = [(log327)/(log310)]
  • Since the 10 is not written
log27 = [(log327)/(log310)]
Write using common logarithms:
log636
  • We're going to use the change of base formula which states that
  • logax = [(logbx)/(logba)]
  • log1036 = [(log636)/(log610)]
  • Since the 10 is not written
log36 = [(log636)/(log610)]
Write using common logarithms:
log452
  • We're going to use the change of base formula which states that
  • logax = [(logbx)/(logba)]
  • log1052 = [(log452)/(log410)]
  • Since the 10 is not written
log52 = [(log452)/(log410)]
Write using common logarithms:
log12100
  • We're going to use the change of base formula which states that
  • logax = [(logbx)/(logba)]
  • log10100 = [(log12100)/(log1210)]
  • Since the 10 is not written
log100 = [(log12100)/(log1210)]

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

Answer

Common Logarithms

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

  • Intro 0:00
  • What are Common Logarithms? 0:10
    • Real World Applications
    • Base Not Written
    • Example: Base 10
  • Equations 1:47
    • Example: Same Base
    • Example: Different Base
  • Inequalities 6:07
    • Multiplying/Dividing Inequality
    • Example: Log Inequality
  • Change of Base 12:45
    • Base 10
    • Example: Change of Base
  • Example 1: Log Equation 15:21
  • Example 2: Common Logs 17:13
  • Example 3: Log Equation 18:22
  • Example 4: Log Inequality 21:52

Transcription: Common Logarithms

Welcome to Educator.com.0000

We are going to continue our discussion of logarithms by talking about special logarithms.0002

And the first one is common logarithms.0008

First of all, what are common logarithms?0011

Common logarithms are logarithms to the base 10; and these are used very frequently in many real-world applications.0013

For example, the scale that is used to measure the magnitude of earthquakes is base 10 scale.0020

And the base 10 is not written: logs in base 10 are just written as follows: log(x).0027

So, if I were discussing log10(7), I could instead just write that as log(7).0037

And it is assumed that, if nothing is written here, then this is base 10--0046

in the same way that, if you took a square root of 4, we know that this really means this;0050

but we don't write the number (because it is so commonly used)--we don't write the number 2.0058

If we are talking about some other root (like the third root--the cube root), then we would write it.0064

The same idea here: if you don't see anything here, you can assume that it is a base 10 log.0068

Now, one application that we are going to have with this is to use logarithms to help us solve exponential equations.0075

In previous lessons, we used exponential expressions to help us solve logarithmic equations.0081

When we had a logarithmic equation with a log on only one side, we converted that to the exponential form.0087

Well, it works the other way as well: there are times when taking an exponential equation0094

and taking the log of the equation can actually help you to solve it; so let's talk about that right now.0098

If we are working with an exponential equation, the previous techniques we used involved having the bases be the same.0108

For example, given something such as an exponential equation 3x + 1 = 94x,0115

we were only able to solve this if we had the same base.0124

And sometimes I could convert it to the same base if I didn't have the same base, because 9 is equal to 32;0131

therefore, I would end up with 3x + 1 = 38x, and then from there, you can solve,0138

because if the bases are equal, then the exponents must be equal.0152

However, getting a little bit more advanced: you are going to run into situations like this: 23x + 4 = 5.0157

These are not the same base, nor can I easily convert them to the same base.0166

In this case, if both sides cannot be written as powers of the same base0169

with an exponential equation, solve the equation by taking the common logarithm of each side.0174

The common logarithm means the base 10 log, which I am just going to write without the 10 here, as is standard.0180

So, log (and that is actually base 10) of 23x + 4 = log(5).0192

Once I have it in this form, then I can solve: you treat this as you would any other equation, which is just by isolating x.0201

Now, let's first rewrite this, using the power property, so that we get rid of this exponent.0215

Let's just go ahead and write this as a coefficient: (3x + 4)log(2) = log(5).0223

I have my variable in here; I want to isolate the x, so I am going to divide both sides by log(2).0235

Keep in mind that these are just numbers--there are no variables in here.0245

log(2) has a specific value; log(5) has a specific value; I haven't found that value, but having my answer in this form is perfectly valid.0250

If you look on your calculator, there is a log button; and that log button, if it just says 'log,' is for the base 10 log,0257

although on some calculators you can specify other bases.0264

This is something you can easily find the value of.0267

So, by doing this, now I have the x much more freed up: I am going to subtract 4 from both sides.0270

And now, all I have to do is divide each side by 3.0281

And instead of writing this as a complex fraction, log(5)/log(2)/3, I can simply remember that this is the same as this.0286

So, if I said log(5)/log(2) times the reciprocal (which would be times 1/3)...I want to get rid of that complex fraction,0312

so I am going to write it like this, in a more simplified-looking form.0320

OK, therefore, I started out where I had an exponential equation where there were different bases.0330

And I couldn't convert them to the same base very easily at all.0336

So, I went ahead and just took the common log of both sides, then used the power property in reverse to get this over here as a coefficient.0339

Next, I divided both sides by log(2), because that is really just a number.0355

And I isolated the x on the left; and everything I have here, I either have a value for, or I can find a value for.0359

The same techniques that we used for solving exponential equations can also be used to solve inequalities--0367

exponential inequalities where we can't get the same base very easily at all.0376

Recall that, when you are multiplying or dividing both sides of an inequality, you need to make sure that you are not working with a negative.0381

And usually, it is obvious: in past lessons, we have known if we are dividing by -3 or multiplying by -3;0391

we need to flip the inequality sign--it is obvious.0396

But you have to be careful with a logarithmic expression, if you are taking the log of some number,0399

that you check and make sure that you are not ending up with a negative,0404

inadvertently, and then having the inequality symbol be in the wrong direction.0408

All right, so let's look at 43x - 5 > 32x - 6.0414

As we did with equations that were exponential equations with different bases, we can do the same technique with this inequality,0426

which is to go ahead and take the common log of both sides.0435

So, this new inequality is still valid, because I did the same thing to both sides of the inequality--0449

the same as if I had added a number to both sides or multiplied both sides by a number.0454

This is still a valid inequality: the relationship between the left and right sides is still holding up.0458

Next, I am going to use the power property to bring this out in front, because my goal is to isolate x.0465

And I can't isolate x when it is up there as a power.0470

3x - 5 times log(4) is greater than...bring this out in front...2x - 6 times log(3).0474

All right, next I am going to divide both sides by log(4).0487

Actually, we need to take one more step: we can't just go ahead and do that.0503

What I need to do is separate this out; I need to use the distributive property to split this out,0506

because what I really have here is 3xlog(4) (that is 3x times log(4)--we can look at this like this--it makes it much easier), minus 5log(4).0514

OK, and this is greater than 2xlog(3) - 6log(3).0540

Now that I have this, and it is split apart, I can go ahead and look at the expressions that do not contain variables.0554

This does not contain a variable, and neither does that; I want those all on the right, and my variable-containing expressions over on the left.0562

I am going to start out by adding 5log(4) to both sides.0572

My next step is going to be to subtract 2xlog(3) from both sides.0587

All right, what I have now is expressions containing variables on the left, and those with constants only are on the right.0605

And again, this is really just a number.0612

Let's go back up here and continue on.0615

And my next step is going to be to factor out x from this left side of the inequality.0618

And this is going to give me an x here, times 3log(4), minus 2log(3), is greater than -6log(3) + 5log(4).0628

So, looking at what I did going from here to here, I just factored out an x.0648

I pulled that x out of here and here, leaving me with this difference: 3log(4) - 2log(3).0653

Now that I have done this, I can isolate the x at last.0662

I just keep x on this side, and I divide (I am going to rewrite this with 5log(4) in the front0664

and the negative second, because it is more standard) by 3log(4) - 2log(3).0673

OK, so in order to solve this exponential inequality that had different bases,0686

I proceeded by first taking the common log of both sides,0692

then isolating the x (which was a little bit complicated).0697

I used the power property, and then I used the distributive property0700

to separate out this 3x and the 5, since this contains a variable and this doesn't; and the same here.0705

Then, I went ahead and factored out an x, and then just divided both sides by this expression.0713

So, I moved the constants to the right, and the variable-containing expressions to the left, and then divided.0724

Now, as I mentioned, you have to make sure that you don't end up dividing by something negative inadvertently.0728

So, at the division step (that was this), let's go ahead and take a look at this.0734

I know that I am OK, because the log of a number greater than 1 is positive.0739

So, if I take the common log of a number greater than 1, it is positive.0744

I know this is positive, and I know that the log of 4 is going to be bigger than the log of 3.0748

And I know that 3 is bigger than 2; so this is going to be larger than this, so I know I am working with something positive;0753

and I don't need to flip the inequality symbol--but that is important to check.0760

The last new concept we are covering in this lesson is going to be change of base.0766

There are times when you have a log given in one base, and you want to change it to another base.0771

Frequently, it is base 10 that you are wanting to change it to, but not always; it could be a different base.0778

And so, this change of base formula allows us to write a given logarithmic expression in a different base.0785

And it allows us to evaluate logarithmic expressions in any base by rewriting it using common logarithms.0792

And as I said, frequently what you want to change the base to is a base that is log base 10.0805

What this right here is, is the original expression.0813

This is actually the original expression; and what we do is divide that original expression by a log to the original base.0820

And what we take the log of is a value that is equal to the new base.0831

So, I take just the original expression and divide it by a log to that base;0835

but for this number, I use the new base that I want.0841

Illustrating this: let's say I had log6(8), and I want to write it as log10.0844

So, I want to write this as log10; I don't need to write the 10 here.0859

This with a new base would be equal to log6(8), the original expression,0870

divided by a log with the original base (base 6), but for the x value right here, I am going to use 10; that 10 is implied.0880

So, if I needed to, for some reason, change this to base 10, this is equivalent to log10(8).0892

You just need to learn this formula and follow it, and know that it is the original expression,0904

divided by a log of the original base; and you are taking the log of the value equal to the base you are looking for.0908

OK, in Example 1, we are looking at an exponential equation where the bases are not the same, and I can't easily get the bases the same.0922

So, I am going to use common logs to solve this.0930

I am going to start out by taking the common log of each side.0933

And I am allowed to do this, as long as I do the same thing to both sides.0938

Once I have it in this form, I want to isolate x; so I need to use the power property to get (2x - 3)log(3) = log(19).0944

To isolate x, I divide both sides by log(x), because my x is here; I want to split that away.0957

Next, I am going to add 3 to both sides, so that the 3 ends up on the right.0967

And then finally, I am going to divide both sides by 2.0980

Let's come up here in the second column: 2x = log(19)/log(3), divided by (2 + 3).0983

And to make this look at little better, we can rewrite this as...actually, the x is now isolated, because we divided by 2;0995

so x equals 1/2, because if I take log(19)/log(3), divided by 2, that would be the same as multiplying by the reciprocal, 1/2.1003

So, I was able to solve this, even though this exponential expression had different bases,1024

by taking the common log of each side, and then isolating x.1028

Write using common logarithms: log7(22).1036

I need to use my change of base formula, which is loga(x) = logb(x), divided by logb(a).1039

And here, a is the new base I want, and b equals the original base--the base that I already have.1050

I want to find an expression that is equivalent to an expression with the original base.1059

OK, log10(22): this is what I want, and I need to find an expression that is equivalent to that.1069

And this would be just the original, log7(22), divided by log with the original base;1080

and then, for a, I am going to use this base that I want.1090

So now, I have written this using common logarithms.1094

OK, again, I have an exponential equation in which there is no common base, and I can't easily get a common base.1104

So, I am going to take the common log of both sides to help me solve this.1111

Now, I need to isolate x; so I need to get the x out of the exponent--I am going to do that by using the power property.1122

And this becomes the coefficient (2x - 6)log(5) =...I am going to bring this out in front...(4x + 3)log(7).1128

Now that I have done this, I am going to write it like this to make it clear what needs to be done.1148

And we need to use the distributive property, because I want to split away these terms that have x in them.1154

I am going to multiply 2x times log(5), and then I am going to multiply -6 times log(5), 4x times log(7),1159

and remember, this is base 10 that we are talking about; and 3 times log10(7).1173

Next, I am going to get all of the variables on the left, and expressions containing constants only on the right.1183

And I am considering the log of 5 to be a constant, because we can find a specific value of that.1188

So, I am going to add 6log(5) to both sides.1192

The next thing I need to do is subtract 4log(7) from both sides, so I can get this on the left, because it does contain a variable.1209

So, 4x - 4x log(7); now I have my constants on the right and variables on the left.1218

The next thing to do: I want to isolate x, so I can factor out an x, because there is an x factor here and one here.1227

I pull that out; I leave behind 2log(5); I pull out the x; I leave 4log(7).1238

And I am almost there; all I need to do now is divide both sides by this expression, and I get x = 3log(7), plus 6...1247

and this is log...that is a 5, not an exponent of 5...all divided by 2log(5) - 4log(7).1262

Again, the technique is to take the common log of both sides, isolate x by first using the power property,1281

then the distributive property, adding and subtracting as needed to bunch all of the expressions and terms1288

containing x's on the left, constants on the right; factor out x, and divide both sides by this expression.1297

And now, I have x isolated, so I have solved for x.1307

This time, we are going to solve an inequality; and it is an inequality involving exponential expressions1314

without a common base, and where I can't easily find a common base.1322

So again, as I did with the exponential equation, I am going to take the common log of each of these.1327

My next step is going to be to isolate the x.1339

All right, so in order to do that, I need to get the x out of being an exponent, so I can use that power property.1346

4x - 5 times log(3) is less than...this becomes...(-3x + 5)log(6).1354

Further separating out the x, I need to use the distributive property: this is going to give me 4x...1365

this is equivalent to this...so 4xlog(3) - 5log(3) is less than -3xlog(6) + 5log(6).1372

As usual, we are going to move the variables to the left and constants to the right.1392

So, I am going to add 5log(3) to both sides; and now I have this x on the right, so I need to move this term to the left by adding 3xlog(6) to both sides.1397

I look on the left; I am still trying to isolate the x, and I see that these two terms have a common factor of x,1438

so I factor that out to leave behind 4log(3) + 3log(6) < 5log(6) + 5log(3).1444

Now, I am going to divide both sides by this expression; and I will have x isolated.1458

Now, I am done; but I just need to double-check about this step, because, whenever you multiply or divide1475

both sides of an inequality, you need to make sure that you are not multiplying or dividing by a negative number,1484

because if you are, you need to reverse the inequality symbol.1490

So, I am looking right here; and I divided both sides by 4log(3) + 3log(6).1494

However, I know that I am OK, because the common log of a number greater than 1 is positive.1501

So, this is positive; and I am adding that to something else positive; so I know that I am OK--that that comes out to be the log of a positive number.1508

That concludes this lesson about common logarithms on Educator.com.1517

Thank you for visiting!1523

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