INSTRUCTORS Carleen Eaton Grant Fraser

Home » Mathematics » Algebra 2 » Logarithms and Logarithmic Functions

Dr. Carleen Eaton

Dr. Carleen Eaton

Logarithms and Logarithmic Functions

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Table of Contents

Section 1: 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
Section 2: 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
Section 3: 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
Section 4: 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
Section 5: 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
Section 6: 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
Section 7: 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
Section 8: 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
Section 9: 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
Section 10: 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
Section 11: 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 (19)

0 answers

Post by Daniel Nie on August 10, 2019

in example 3 you wrote -3+2=1

1 answer

Last reply by: DJ Sai
Mon Sep 17, 2018 9:59 PM

Post by John Stedge on June 15, 2018

at 12:42 you spelled asymptote wrong

1 answer

Last reply by: Dr Carleen Eaton
Sat Nov 7, 2015 6:06 PM

Post by Peter Ke on October 21, 2015

Hi, what is a asymptote? I search it on google and I don't really get it. Can you explain it in an easier way?

0 answers

Post by julius mogyorossy on June 22, 2013

The inverse property for logs is like, 2, times 4, divided by 4, = 2.

0 answers

Post by julius mogyorossy on June 10, 2013

Dr. Eaton, again I don't know why you say x'2-x-12 is y, when you set the right side of the equation to 0, what I call y, it seems you are making it = x. I think the quadratic equation should not be called y or x, but the determinator, something like that. It just confused me, when you set y to 0, then you said x'2-x-12 is y. I am not sure why you talk about y at all when showing ex-4, it does not seem that y has anything to do with that problem, or why you show a quadratic equation graph line, it seems you should have only showed a x dimension line graph. That problem is about finding values for x that make the inequality true, nothing to do with y. I kept thinking there was something I was not seeing. Again the quadratic equation factored said that part of the solution set was x>-3, not, x<-3, as you said, what you would call an invalid solution, I guess, and it said x>4 is possibly part of the solution set, which is true. Yes, you talked about y, but it really seems it is just about x and x, I know you said one of the x's is a y, but really for this problem it just seems like it is about x and x, it seems that is the better way to see it so as not confuse it with a regular quadratic-inequality problem. It really seems the y values for this problem are irrelevant. The equation is set to 0 not to find the roots, the y's=0, but to find the solutions that also do not violate the rule to not take the root of 0 or a #<0, isn't that the case. For on kind of problem you set it to 0 to find the roots, for another to make sure not to take the log of 0 or a #<0. It seems you unnecessarily factored x'2-9 and x+3 twice, the first time when you combined them into a quadratic equation, killing two birds with one stone, so to speak, then again separately, unnecessarily it seemed, is this true.

1 answer

Last reply by: Dr Carleen Eaton
Sun Mar 11, 2012 7:16 PM

Post by Jeff Mitchell on March 7, 2012

Dr Carleen,
In example IV, you where checking for exclusions and said
(x+3)(x-3)>0
X+3>0
X>-3
X-3>0
X>3
and you indicated -3 is not a valid solution
but wouldn't it be true that if x=-4;
(-4+3)(-4-3)=(-1)(-7)>0 which is valid?

1 answer

Last reply by: Dr Carleen Eaton
Mon Nov 7, 2011 8:49 PM

Post by Jonathan Taylor on November 6, 2011

Dr Carleen would u explain how either u subtract or add x+4>9(equalities)
x=5

1 answer

Last reply by: Dr Carleen Eaton
Fri Jun 10, 2011 1:01 AM

Post by Manuel Gonzalez on June 7, 2011

aren't roots the opposite of exponents?

0 answers

Post by Edgar Rariton on March 6, 2011

The definition of a logarithm in this video begins with "First the restrictions...". That's one way to do it, but perhaps explaining that logarithms are the "opposite" of exponentials would have been better.

0 answers

Post by Wade Sias on January 27, 2011

There needs to be more complex problems like
lnx + lnx(x-3) <= ln4
I feel the examples are to easy and I am struggling with these longer problems.

1 answer

Last reply by: Suhani Pant
Sat Jul 27, 2013 12:16 PM

Post by Dr Carleen Eaton on August 20, 2010

Correction to Example III at 38:42
When checking the potential solution x = 3, I should have written the original equation as log base6(x squared- 6) NOT
log base6 (x squared - 3). With the correct equation, substituting x = 3 would give:
logbase6(3 squared - 3)
logbase6 (9-3)
logbase6(3)
Therefore, x = 3 is a valid solution.
I apologize for any confusion my error may have caused.

Logarithms and Logarithmic Functions

  • Remember that a logarithm is just an exponent.
  • Understand that the log function and the exponential function are inverses of each other.
  • Use this to solve problems.
  • Solve logarithmic equations with the same base by equating the expressions whose logarithms have been equated.
  • Solve logarithmic inequalities with the same base by applying the same inequality to the expressions whose logarithms have been compared.
  • In solving logarithmic equations or inequalities, always check for extraneous solutions – values which result in taking the logarithm of a non-positive value in the original equation or inequality. Exclude such values from the solution set.

Logarithms and Logarithmic Functions

Solve log4(x4 − 17) = 3
  • Recall that logbx = y is the same as by = x
  • b = 4
  • y = 3
  • x = x4 − 17
  • Rewrite into exponential form.
  • 43 = x4 − 17
  • 64 = x4 − 17
  • Solve
  • x4 = 81
x = 3
Solve log4(x4 − 192) = 3
  • Recall that logbx = y is the same as by = x
  • b = 4
  • y = 3
  • x = x4 − 192
  • Rewrite into exponential form.
  • 43 = x4 − 192
  • 64 = x4 − 192
  • Solve
  • x4 = 256
x = 4
Solve log4(x4 − 369) = 4
  • Recall that logbx = y is the same as by = x
  • b = 4
  • y = 4
  • x = x4 − 369
  • Rewrite into exponential form.
  • 44 = x4 − 369
  • 256 = x4 − 369
  • Solve
  • x4 = 625
x = 5
Solve log2(x3 − 713) = 4
  • Recall that logbx = y is the same as by = x
  • b = 2
  • y = 4
  • x = x3 − 713
  • Rewrite into exponential form.
  • 24 = x3 − 713
  • 16 = x3 − 713
  • Solve
  • x3 = 729
x = 9
Solve log5(x2 − 12) = log5(x)
  • Notice that since the bases are the same, you can use the following property
  • logbx = logby; then x = y
  • log5(x2 − 12) = log5(x)
  • x2 − 12 = x
  • x2 − x − 12 = 0
  • Factor
  • x2 − x − 12 = (x − )(x + )
  • x2 − x − 12 = (x − 4)(x + 3) = 0
  • Solve using the Zero Product Property
  • x − 4 = 0;x + 3 = 0
  • x = 4;x = − 3
  • Check Solutions
  •  x=4x=-3
    log5(x2 − 12) = log5(x) log5(x2 − 12) = log5(x)
     log5(42 − 12) = log5(4)log5(( − 3)2 − 12) = log5( − 3)
     log5(4) = log5(4)log5( − 3) = log5( − 3)
     x = 4 is validNot valid, you cannot have negative logarithms.
x = 4
Solve log5(x2 + 4) = log5( − 5x)
  • Notice that since the bases are the same, you can use the following property
  • logbx = logby; then x = y
  • log5(x2 + 4) = log5( − 5x)
  • x2 + 4 = − 5x
  • x2 + 5x + 4 = 0
  • Factor
  • x2 + 5x + 4 = (x + )(x + )
  • x2 + 5x + 4 = (x + 1)(x + 4) = 0
  • Solve using the Zero Product Property
  • x + 1 = 0;x + 4 = 0
  • x = − 1;x = − 4
  • Check Solutions
  •  x=-1x=-4
    log5(x2 + 4) = log5( − 5x)log5(x2 + 4) = log5( − 5x)log5(x2 + 4) = log5( − 5x)
     log5(( − 1)2 + 4) = log5( − 5( − 1))log5(( − 4)2 + 4) = log5( − 5( − 4))
     log5(5) = log5(5)log5(20) = log5(20)
     x=-1 is validx=-4 is valid
x = − 1; x = − 4
Solve log3(5x − 3) < 3
  • Recall that logbx = y can be written in exponential form as by = x.
  • Don't forget that there's always a restriction when working with logs, namely
  • logbx < y; then 0 < x < by
  • Solve
  • 0 < x < by
  • 0 < 5x − 3 < 33
  • 0 < 5x − 3 < 27
  • 3 < 5x < 30
[3/5] < x < 6
Solve log12(9x − 18) < 2
  • Recall that logbx = y can be written in exponential form as by = x.
  • Don't forget that there's always a restriction when working with logs, namely
  • logbx < y; then 0 < x < by
  • Solve
  • 0 < x < by
  • 0 < 9x − 18 < 122
  • 0 < 9x − 18 < 144
  • 18 < 9x < 162
2 < x < 18
Solve log7(9x − 18) < 2
  • Recall that logbx = y can be written in exponential form as by = x.
  • Don't forget that there's always a restriction when working with logs, namely
  • logbx < y; then 0 < x < by
  • Solve
  • 0 < x < by
  • 0 < 10x + 9 < 72
  • 0 < 10x + 9 < 49
  • − 9 < 10x < 40
− [9/10] < x < 4
Solve log3(8 + 3x) < log3(x2 − 2)
  • Since the base of the exponents is the same, we can take what is inside the parenthesis outside
  • 8 + 3x < x2 − 2
  • Move everything to one side of the inequality
  • 0 < x2 − 3x − 10x
  • Factor
  • 0 = (x − 5)(x + 2)
  • Solve using the Zero Product Property
  • x − 5 = 0
    x + 2 = 0
  • x = 5
    x = − 2
  • In this case, x <− 2 and x > 5 Now check restrictions by the logs
  • log3(8 + 3x) < log3(x2 − 2)
  • 8 + 3x > 0 3x >− 8 x >− [8/3]
  • x2 − 2 > 0 x2> 2 x >√2
  • Notice how there's two situations here. while it is true that x can be less than − 2, the restriction
  • in the log forces this value to be grater than − [8/3]. Also, the second restriction, that x be greater than √2
  • is taken care by the fact that x has to be greater than 5. Therefore, putting this together the solution is
− [8/3] < x <− 2 and x > 5

*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

Logarithms and Logarithmic Functions

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 Logarithms? 0:08
    • Restrictions
    • Written Form
    • Logarithms are Exponents
    • Example: Logarithms
  • Logarithmic Functions 5:14
    • Same Restrictions
    • Inverses
    • Example: Logarithmic Function
  • Graph of the Logarithmic Function 9:20
    • Example: Using Table
  • Properties 15:09
    • Continuous and One to One
    • Domain
    • Range
    • Y-Axis is Asymptote
    • X Intercept
  • Inverse Property 16:57
    • Compositions of Functions
  • Equations 18:30
    • Example: Logarithmic Equation
  • Inequalities 20:36
    • Properties
    • Example: Logarithmic Inequality
  • Equations with Logarithms on Both Sides 24:43
    • Property
    • Example: Both Sides
  • Inequalities with Logarithms on Both Sides 26:52
    • Property
    • Example: Both Sides
  • 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

Transcription: Logarithms and Logarithmic Functions

Welcome to Educator.com.0000

Today, we are going to talk about logarithms and logarithmic functions, beginning with the definition.0003

What are logarithms? First, the restrictions: x cannot be 0, and b needs to be a positive number, but not equal to 1.0011

So, the logarithm of x to the base b is written as follows: logb(x).0021

Let's start out with logb(x) = y: in this case, this log is defined to be the exponent y, which would satisfy this equation.0029

Over here, I am talking about the same base: the base b here is the base b right here.0042

We worked with exponents already; and logarithms are actually just exponents.0050

When you use logarithmic notation, you are just writing an exponential expression in a different way.0055

The logb(x) = y is defined as the exponent y that, when you raise b to that power, will give you x back.0062

So, you need to be able to get comfortable with going back and forth between the logarithmic notation and the exponential expression.0076

These two statements are actually inverses of each other; and we will talk more about that relationship in a little while.0084

But we are just starting out now, to get used to the idea of converting back and forth.0090

And the reason that you need to be able to convert back and forth and understand the relationship between these two0094

is: exponential expressions can help you solve logarithmic equations, and logarithmic expressions can help you solve exponential equations.0098

Given log4(16) = 2, you can rewrite that into the exponential expression.0110

At first, you might need to just stop and analyze the various components.0118

The base is 4; here, x is 16, and y is 2; so I am going to rewrite that: the base remains the same:0123

4 to some power y (here it is 2) equals 16; and we know that that is true--that 42 is 16.0132

You can also move from the exponential equation into the logarithmic equation.0140

Looking at an example where you are starting out with 23 = 8:0145

the base here is 2; y = 3 (that is the power you are raising the base to); and x = 8.0152

Now, I am going to write this as a logarithmic equation: the base is still 2; x is 8; and y is 3; so log2(8) = 3.0164

We can also use this for slightly more complicated situations, such as log5(1/125) = -3.0183

But it is the same idea, because I still have the base, 5, and I know that this is y, so 5-3 equals 1/125.0194

And I know that that is true, because 5-3 would be the same as 1/53, which is 1/125; so that holds up.0205

I can use this to evaluate logarithmic expressions when I am trying to solve.0221

Now, here I gave you all of the pieces; you already had all of the numbers, and it was just rewriting them a different way.0226

Let's look at a situation where we are actually trying to find a value.0232

You are given log3(81) = y.0236

I have the logarithmic expression; but I can solve this more easily if I rewrite it as an exponential equation.0246

I have this base, 3; and what this is saying is that a logarithmic expression is defined to be the exponent y satisfying this relationship.0254

So, if I take the base, and I raise it to the y power, I am going to get 81.0266

I just need to figure out what I would need to raise 3 to, to get 81.0276

3 squared is 9; 3 cubed is 27; 3 to the fourth is 81; therefore, y equals 4, because 34 is 81.0282

This shows you how being able to convert between the logarithmic equation and the exponential equation can help you to solve either one of them.0296

OK, looking on at logarithmic functions: a logarithmic function is a function of the form f(x) = logb(x).0310

We just introduced this idea of logarithms; now we are talking about logarithmic functions, where b is greater than 0 but not equal to 1.0321

And you will call that these are the same restrictions that we had when talking about exponential equations.0330

And this would be for the same reasons as discussed in that lecture, because again, logarithms are simply exponents.0337

As I mentioned in a previous slide, these two are inverses; so f(x) = logb(x)0347

is the inverse of the exponential equation, expression, or function g(x) = b(x).0355

So, f(x) = logb(x) is the inverse of g(x) = bx.0363

And this is an important relationship, because it helps us to solve the equations that we will be working with shortly.0374

Let's just take an example to make this more concrete.0380

Let's let f(x) equal log2(x): the inverse of that would then be g(x) = that same base, 2, raised to the x power.0385

Let's look at some values for f(x): if x is 1, what is y?0400

Well, think about what this is saying: this is saying log2(x) equals some value of y.0408

Rewriting that as a related exponential expression, this is telling me that, when I take 2 and I raise it to the y power, I am going to get x back.0415

What I want to figure out here is: 2 to some power, y, equals 1; what would y have to be?0425

y would have to be 0; therefore, 20 = 1 satisfies this: x is 1 when y is 0.0432

2 to some power y equals 2; what would y have to be? It would have to be 1.0444

How about 4? 2 to some power y equals 4; 22 = 4; therefore y = 2.0451

Let's let x be 8: 2 to some power y equals 8: well, 23 is 8; therefore, y = 3.0461

All right, so that is f(x); now let's look at g(x).0469

If f(x)...this is f(x), but here I am calling it y; now let's look at g(x)...and g(x) are inverses,0475

then what I am going to expect is that the domain of f(x) is going to be the range of g(x),0484

and the range of f(x) is going to be the domain of g(x).0490

So, I am going to go ahead and take these values right here that are the range of f(x);0493

and I am going to use them as the domain of g(x), and see if I get these values back.0500

So, when x is 0, y is 2 to the 0 power, or 1; when x is 1, y is 2; when x is 2, y is 22, is 4.0509

When x is 3, y is 23: it is 8; and I look, and the domain here is equal to the range; the range here is equal to the domain.0529

Now, that doesn't prove anything: it just shows that this one example holds up.0540

But our finding was, as expected, that if f(x) and g(x) are inverses, I do expect the domain of one to be the range of the other,0544

and the range of this one to be the domain of that one.0552

Looking at the graphs of these functions, we can use a table of values to graph a logarithmic function,0560

just as we have used tables of values to evaluate functions earlier in the course, including exponential functions.0566

So, we already started a table for f(x) = log2(x), and for g(x) = 2x, the inverse.0572

So, let's keep going with those, but add on some values.0584

Recall that I said, if f(x) = log2(x), then if I take 2y, I am going to get x.0591

I am rewriting this in exponential form to make it easier for me to find y, because it is difficult to find f(x), or y, when it is in this form.0600

Recall that I said that, when x is 1, then what this is saying is that 2y = 1.0610

And I said that y must be 0; when x is 2, 2y = 2, so y must be 1.0617

And I went on and did a couple of other values, and I did 8: I said 2y = 8; therefore, y had to equal 3.0628

Now, let's add some values: let's add some fractions to get a better idea of what this graph is doing as x becomes small, as it gets close to 0.0639

When x is 1/2, that is telling me 2y = 1/2.0651

The way I would get that is if I took 2-1; I would get 1/2; therefore, y is -1.0657

1/4: 2y = 1/4--if I took 2-2, I would get 1/4, so y is -2.0668

1/8: using that same logic, 23 is 8, but 2-3 would be 1/8, so I am going to make that -3.0678

And then, I am going to graph these values.0688

When x is 1, y is 0; when x is 2, y is 1; when x is 4, y is 2; and when x is way out here at 8, y is 3.0691

So, the general shape is just going up like this.0705

Small values: when x is getting smaller and smaller, what is going to happen with y as x is approaching 0?0709

When x is 1/2, y is -1; when x is 1/4, y is -2; when x is 1/8, y is -3.0719

And what I can see is happening here is that the y-axis is a vertical asymptote.0730

And we talked earlier about the graphs of exponential functions: we saw that the x-axis formed a horizontal asymptote.0740

Here, the y-axis is an asymptote.0746

All right, let's go ahead and look at the graph of the inverse.0760

And that is going to be very simple, because I know that, since this is the inverse, all I have to do0762

is take the domain and make that the range, and then I take the range and make that the domain.0770

And I am going to check and make sure I have all of these values correctly matched up: 1 and 2...yes, I do.0787

Then, I am going to go ahead and graph them: when x is 0, g(x) is 1 (this is f(x)).0799

When x is 1, g(x) is 2; when x is 3, g(x) is way up here at 8, about here.0805

When x is -1, y is 1/2; -2, 1/4; at -3, it is 1/8.0815

Now, I know that I have an exponential function here that I am graphing.0827

So, as expected, the x-axis is an asymptote for g(x), written in this exponential form.0833

So, looking at what this is saying: the vertical asymptote here is at x = 0, so x will never cross this axis.0855

It will never become negative for f(x), and that makes sense.0869

x can never be negative, because there is no value of y that I can take...2 to some value...and get a negative back.0873

So, x cannot be negative, because there is no possible value of y that would turn 2 into a negative number.0884

Therefore, the domain of f(x) is restricted to values greater than 0 (to positive values).0890

Now, since this is the inverse, and the range of this g(x) is going to be the same as the domain of that,0898

that means that the range of g(x) is going to just be all positive numbers.0904

To sum up properties: the graph of a logarithmic function f(x) shows that f(x) is continuous and one-to-one.0910

There are no gaps; there are no discontinuities; also, you can take that graph that we just did and try the vertical line test.0917

No matter where you drew a vertical line, it will only cross the curve of f(x) once.0924

Therefore, this is a function; there is a one-to-one relationship between the values of x and the values of y.0930

We just discussed why the domain of f(x) must be all positive real numbers.0936

The domain cannot include a negative number, because there would not be any value for y that would give you a negative number.0942

You cannot end up with a negative value for x.0953

The range, however, is all real numbers; so y can be a negative number--I can have -2 here, or something.0956

We also saw that the y-axis is an asymptote, and that the graph is going to approach that axis, but it is never actually going to reach it.0962

And finally, just sketching this back out again, the graph of f(x) looks like this.0973

This is the graph of f(x), and I used log2(x) for that.0990

And the y-intercept was at (1,0), and illustrated here, the y-axis is an asymptote.0999

The domain is only positive numbers; however, the range is all real numbers.1007

Since f(x) equals bx, and g(x) is logb(x), and they are inverses of each other,1017

we can end up with the identity function when we use a composite function, or composition of functions.1023

Recall that, when we talked about composition of functions, we ended up with something like this: f composed with g equals f(g(x).1030

Applying that up here, that is going to give me f(logb(x)) = b...and this is going to be my x;1044

so for x, I am going to insert this: logb(x).1055

And since these are inverses, this two essentially cancel each other out; and I am just going to end up with my x back (the identity function).1058

g composed with f should do the same thing: this is g(f(x)) = g of f(x), which is bx.1070

g here is logb(x), so I am going to take logb, and for x right here, I am going to substitute in bx.1080

And this will work as an identity function, giving me x back.1092

And the inverse property is going to be very helpful to us, as we work on solving equations involving logarithms and exponents.1099

Let's take a look at some methods for solving equations, starting with just very simple ones, and then advancing to more complex equations.1111

Logarithmic equations are defined as those that contain one or more logarithms with a variable in them.1119

The definition can be used to solve simple logarithmic equations.1125

First, recall what the definition of the logarithm is: logb(x) = y if this base, b, when raised to the y power, generates x.1130

This definition can be used to solve very simple logarithmic equations where there is a log only on one side of the equation.1142

By log, I mean a log containing a variable.1149

logb(x + 4) = 3: recall that I said that, if you have a logarithm, you can often use the exponential form to help you solve the equation.1154

And later on, we will see that, if you have the exponential form, you can use the log to help you solve equations.1166

So, thinking this out: I know that I have the base equal to 2 and that x = x + 4 and y = 3.1173

So, I can rewrite this in this form: my base 2, to the third power, equals x, which is x + 4.1180

Now, I have something I can solve: 23 is 8; 8 = x + 4.1190

All I have to do is subtract 4 from both sides, and I get x = 4.1195

Now, you have to be careful when you are working with logs, because we can't take the log of a negative number.1202

So, I am going to just check back in my original here and see...1208

If I put this 4 in here, I am going to get log2(4 + 4), which is log2(8).1213

And that is fine, because that is a positive number.1223

I went ahead and solved this, and then I just double-checked that I ended up with a value that is allowed.1226

Logarithmic inequalities are a similar idea: these are inequalities that involve logarithms.1237

And if b is greater than 1, and x is a positive number, and I have logb(x) > y...1242

(again, we are talking about just a very simple situation, where there is a log on only one side), then x is greater than by.1251

Look at what we did here: we went from the log form to the exponential form.1260

Recall that, with equations, it would look like this: we are doing this same thing, only we are doing it with inequalities.1270

And this relationship still holds up: if the logb(x) > y, then x must be greater than by.1280

It is a little more complicated with less than: so let's just start out talking about greater than, and illustrating it with an example.1294

I am starting out with log3(x + 4) > 2.1300

I know that, if this is true, then I can convert it to this form to solve, because this relationship will hold up.1312

What this is telling me is that, if I take my base equal to 3, and x is equal to (x + 4), and y is equal to 2, I can convert it to this form.1322

So, x is right here: x + 4 is greater than my base b, 3, raised to the second power.1338

Now I can go ahead and solve: x + 4 > 9, so x > 5.1350

I always have to be careful with logs, and make sure that I don't end up taking the log of a negative number.1356

So, log3(x + 4): if I put 5 in here, or slightly more than 5 (x is greater than 5)...say 5.1, I am going to get, say, 9.1, which is positive.1364

So, I am not worried about taking the log of a negative number, because in order to get a negative number,1377

you would have to have something that would be less than 4, and that is not going to occur here.1382

So, that is fine: it is more complicated with less than, and let's look at why.1387

For less than, let's say I had log4(x - 1) < 3.1391

I can't just come up with some number, some value, "x is less than 2," because the problem is:1399

then I could get into very, very small numbers--very negative numbers--1405

where I could end up taking the log of a negative number, which we are not doing.1410

I have to put a restriction on the other side; I have to make sure that x is greater than 0.1416

So, when I convert to this form, instead of just x < by, I need to say, "but it is also greater than 0."1422

So, the base here is 4; I have my 0 here; x in this case is x - 1; the base is 4; and y is 3.1436

So, looking at this: the base is 4; x is x - 1; and y is equal to 3.1453

Therefore, 0 is less than x - 1; 4 times 4 is 16, times 4 is 64.1460

I am going to add a 1 to both sides; and this is going to give me that x is greater than 1, but less than 65.1470

It has put a restriction, a lower limit, on this.1479

OK, we just discussed solving equations with a logarithm on one side.1482

In order to solve equations with logarithms on both sides, we use the following property.1486

If the base is greater than 0, but not equal to 1, then if you have a logarithmic equation1491

with a log on each side, and whose bases are the same, then x must equal y.1497

So, the restriction here is that the bases need to be the same.1505

And you might remember back to working with exponential equations: we said that, if you have an exponential equation,1507

and the bases are the same, then the exponents must be equal.1513

It is similar logic here, which is not surprising, since this is just a notation for working with exponents.1516

For example, log4(3x - 1) = log4(x + 5).1524

Since these bases are the same, then in order for this equation to be valid, 3x - 1 has to equal x + 5.1533

This leaves me with a linear equation that I can easily solve.1544

First, I am going to add a 1 to both sides to get 3x = x + 6; then I am going to subtract an x from each side to get 2x = 6.1547

Then I will divide both sides by 2 to get x = 3.1559

Remember, when working with logs, it is very important to check and make sure that you have a valid solution,1563

because you can't take the log of something that is negative.1567

So, I need to make sure that this expression is not going to end up negative, and this expression is not going to end up negative.1572

So, I am going to plug in this value and see what happens.1579

For this first one, I am going to get log4(9 - 1), which is log4(8); that is valid.1584

Now, this should be the same in here, since these two are equivalent; but we will just double-check it anyway.1592

log4(x + 5) would be log4(3 + 5), or log4(8).1597

And since that is taking a log of a positive number, that is allowed, and this is a valid solution.1604

When working with inequalities with logarithms on both sides, we can solve these inequalities with the following formula.1612

If b is greater than 1, then if you have a log with a certain base, logb(x) > logb(y),1619

then this relationship is true if and only if x is greater than y.1630

The relationship between these two is maintained as long as x is greater than y; so that is a given.1634

And logb(x) is less than logb(y) if and only if x is less than y.1642

This is similar to the logic that we use when solving logarithmic equations with one log on each side.1648

We are doing the same idea, but with inequalities.1655

You need to make sure that you exclude solutions that would require taking the log of a number less than or equal to 0 in the original inequality.1658

So, we are going to look for excluded values at the end, and make sure that we remove those from the solution set.1666

Solutions to the inequality need to actually make the inequality valid.1674

And they need to be not part of the excluded values.1681

I will illustrate that right now: log5(3x + 2) ≥ log5(x - 4).1685

Since the bases are the same, I know that this needs to be greater than or equal to this expression.1698

So, 3x + 2 ≥ x - 4.1706

Therefore, just solving this inequality is going to give me 3x ≥ x - 6; 3x - x ≥ -6; 2x ≥ -6.1716

Divide both sides by 2; I have x ≥ -3.1734

And it might be tempting to stop there; but I need to go back and look at the original.1738

I need to make sure that I am taking only values (in my solution set) that are not excluded.1745

When we were working with equations, it was simple: we would get something like x = 2.1752

And then, we just had to plug it in here and make sure that that was valid; we were fine.1755

Here, we have a whole solution set; so I have to use inequalities to find excluded values,1760

or to find what x must be to end up with a valid solution set.1767

So, I am going to look at log5(3x + 2): for this to be valid, I need for this in here to be greater than 0.1776

All right, that would give me 3x > -2, or x > -2/3.1795

So, this log right here will be valid, as long as x is greater than -2/3.1808

If it is smaller than that, I will end up having an excluded value.1819

Let's look at this log: log5(x - 4): in order for this to be valid, I need to have x - 4 be greater than 0.1825

That means that x would have to be greater than 4.1838

If x is some value less than 4, I will end up taking the log of a negative number, or 0; that is not allowed, so that would not be a valid solution.1840

Now, look at my solution set: my solution set says that x has to be greater than or equal to -3.1849

But that would encompass values that are too small--excluded values--values that are not allowed, like...1855

I could end up with 0, which would then make this -4, and then I would be taking the log of a negative number.1866

So, I have to have a solution set that meets this criteria, this, and this--the most restrictive set.1872

x needs to be greater than or equal to -3, greater than -2/3, and greater than 4.1880

So, I need to go with this: x > 4--that is the solution set.1886

So, when there is an inequality with a logarithm on both sides with the same base,1892

you put this expression on the left side, keep the inequality the same, put this expression on the right, and solve.1898

Then, find excluded values, and make sure that your solution set does not include excluded values.1907

All right, in the first example, we have a logarithmic equation that only has a log on one side.1914

And recall that we can use the definition of logarithms to solve this.1919

logb(x) = y if the base, raised to the y power, equals x.1923

Therefore, I can rewrite this in this form...so, just rewriting it as it is, here the base is equal to 2;1930

x is equal to x3 + 3; and y is equal to 7.1945

So, writing it in this form would give me 27 = x3 + 3.1950

Let's figure out what 2 to the seventh power is: 23 is 8, times 2 is 16, times 2 is 32, times 2 is 64, and then times 2 is going to give me 128.1955

So, this is 2 to the third, fourth, fifth, sixth, seventh: so 27 is going to be 128.1971

All right, I am rewriting this as 128 = x3 + 3.1979

I am going to subtract 3 from both sides: x3 = 125.1986

The cube root of x3 is x, and the cube root of 125 is 5, because 5 times 5 is 25, times 5 is 125.1994

Now, I also have to make sure I don't end up with a negative value inside the log.2004

So, I am going to check this solution to make sure it is valid.2008

x base 2...I am going back to the original...x cubed plus 3...I need to make sure that this is not negative when I use the solution.2011

log2(53 + 3)...well, I know that this is going to be positive; so that is fine--this is a valid solution.2019

All right, log2(2x - 2) < 4--now I am working with an inequality that has a log on only one side.2034

And I am going to go back and think about my definition of logs--that logb(x) = y if by = x.2045

And by rewriting this using the exponential form, I can much more easily solve it.2057

Now, I am working with less than; so I need to recall that, if logb(x) < y, then x is greater than 0, but less than by.2067

If I didn't put this restriction here, I could end up with a value that is too small, and that would be excluded.2090

All right, so rewriting this in this form: I am going to have the 0 here, and x is greater than that.2100

x is 2x - 2...is less than the base, which is 2, raised to 4.2107

I am rewriting this in an exponential form, but making sure that I have this restriction, since we are working with less than.2122

So, 0 is less than 2x - 2; 2 times 2 is 4, times 2 is 8, times 2 is 16.2127

Now, I am going to add 2 to both sides to get 2x is greater than 2, but less than 18;2136

and then divide both sides by 2 to give me x is greater than 1 and less than 9.2143

So, I solved this by using this property of logarithmic inequalities, and making sure that I had the 0, since I am working with less than--2150

that I had the restriction that this expression is greater than 0,2160

so that this in here (what goes inside for the log) does not end up being negative.2165

I don't need to worry about that with greater than.2173

This time, we are going to be solving a logarithmic equation in which there is one log on each side of the equation.2177

Recall that we talked about the property: if there are two logs with the same base, logb(x) = logb(y)--2183

the same bases--then for this equation to be valid, x has to equal y.2193

So, I have log6(x2 - 6) = log6(x).2199

Since these are both base 6, I can just say, "OK, x2 - 6 = x."2206

This is just a quadratic equation: I move the x to the left, and I am going to solve by factoring, just as I would another quadratic equation.2214

This is x + a factor, and then x minus a factor of 6, equals 0.2225

Factors of 6 are 1 and 6, and 2 and 3; and I need them to add up to -1.2232

So, -3 + 2 = -1; I put the 2 here and the 3 here.2238

Using the zero product property, I can solve this, because x + 2 = 0 and x - 3 could equal 0.2247

And either way, this is going to become 0.2255

x = -2 for this; and another solution is x = 3.2259

Now, I have two possible solutions; I need to check these.2263

Let's go up here and look at this, with the -2.2273

I have log6(x2 - 6); if x = -2, then I am going to end up with log6(4 - 6), or log6(-2).2277

That is not valid; I could have also just looked up here and said, "OK, if x equals -2, I would be taking log6(-2)."2292

So, that is not valid; the solution is not valid.2299

Let's try x = 3: well, if x equals 3, and I take the log base 6 of 3, that is OK.2308

Let's check this one out: log6(x2 - 6): log base 6...and we are letting x equal 3 here.2314

of 3 squared minus 6; so that is log6(9 - 6), or log6(3), which is positive.2329

That is allowable; log6(3) is allowable; so these are both allowable.2339

Therefore, the solution is simply x = 3.2344

We came up with two solutions; one was extraneous; we checked and found that we have one valid solution, which is x = 3.2351

Example 4: Solve this inequality that involves a logarithm on each side of the inequality.2360

And I am going to recall that, if the bases are the same (which they are), then logb(x) > logb(y) only if x > y.2367

x must be greater than y; that relationship has to hold up.2381

I am just going to go ahead and look at what I have in here, which is x2 - 9 > x + 3.2385

And I am going to move the 3 to the right by subtracting a 3 from both sides.2394

So, x2 - 9 - 3 > x; so x2 - 12 > x.2401

I am going to subtract an x from both sides; it is going to give me x2 - x - 12 > 0 (the 0 will be left behind).2410

So, this gets into material that we learned earlier on in the course; and it is a little bit conceptually complex.2422

But if you just think it out, you can solve this.2428

Let's factor this out; that is always a good first step.2433

This gives me (x - 4) (x + 3), because the outer terms are -3x - 4x; the inner terms will give me -x; and -4 times 3 is -12.2439

If I wanted to graph this out, I could say, "OK, (x - 4)(x + 3)...let's turn it into the corresponding equation and find the roots."2456

x - 4 = 0, so x = 4; I am finding the roots of this corresponding quadratic equation, just as we did earlier in the course when we were solving quadratic inequalities.2465

Also, x - 3 = 0 would satisfy this equation if x = 3.2477

Actually, that is x + 3 = 0, so x = -3.2487

What this is telling me is that the corresponding quadratic equation has roots at -3 and 4; the zeroes are there.2493

Something else I know is that the leading coefficient here is positive; so this is going to be a parabola that faces upward.2505

And this is really all I need to solve this.2515

And then, I need to go back to my inequality and say, "OK, if this graph looks like this,2518

and what I want is this function, the y-values, to be greater than 0, where are those going to be?"2523

The graph crosses the x-axis right here, at -3: when x is -3, y is 0.2533

For all values of x that are more negative than -3, y is positive.2540

So, for this portion of the graph, when x is less than -3, y is greater than 0, which is what I want.2545

I look over here, and the graph crosses the x-axis at x = 4; so for all values of x that are greater than 4, y is greater than 0.2560

Therefore, in order to satisfy this inequality, x can be less than -3, or x can be greater than 4.2574

Now, I can't stop there and say that I have solve this inequality, because the problem is2587

that I need to make sure that I am not dealing with excluded values.2591

So, let's look at what the excluded values are going to be.2595

If I have log7(x2 - 9), I need for this x2 - 9 to be greater than 0.2597

So, if I factor that, I am going to get (x + 3) (x - 3) > 0.2609

In order for (let's rewrite this a little bit better)...(x + 3) (x - 3) needs to be greater than 0.2620

Therefore, x + 3 needs to be greater than 0, so x needs to be greater than -3.2633

And x - 3 needs to be greater than 0; so x needs to be greater than 3.2639

This is a restriction, and this is a restriction, that if I don't meet these restrictions in my solution set,2647

I am going to end up with excluded values, and it is going to be invalid.2658

So, I am going to look at my solution set: x > 4 meets these criteria, that x > -3 and x > 3.2662

And I don't need to worry about this one, because I already have that factor covered right here: x + 3 > 0.2670

This meets the criteria: it doesn't include any excluded values.2675

However, when I look at x < -3, it doesn't meet all of the criteria, because here, this says x has to be greater than -3.2679

If I take a value, -4, it is not going to meet this criteria; it is not going to meet this criteria, either (that x needs to be greater than 3).2689

Therefore, this is not valid; and my solution is just going to be x > 4,2698

because it solves the inequality--it satisfies the inequality--and it doesn't include excluded values.2705

That one was pretty difficult: first you had to realize that, with the same bases, then you could just take this expression2713

and say that it is greater than this expression and go about solving the inequality.2721

That was a little bit tricky, because then you had to think about what this meant,2726

solve this quadratic inequality, and then realize that x < 3, or values of x that are greater than 4,2730

would satisfy the inequality, but there were some excluded values as part of this solution set.2736

So, we had to just go with x > 4.2742

That concludes this lesson of Educator.com on logarithmic equations and inequalities; and thanks for visiting!2746

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