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INSTRUCTORS Carleen Eaton Grant Fraser
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Dr. Carleen Eaton

Dr. Carleen Eaton

Properties of 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 (10)

1 answer

Last reply by: Dr Carleen Eaton
Thu Mar 27, 2014 6:58 PM

Post by Christopher Lee on January 22, 2014

In the 4th step for example 1, you're missing a parentheses at the end.

1 answer

Last reply by: Dr Carleen Eaton
Thu Mar 27, 2014 6:57 PM

Post by Christopher Lee on January 22, 2014

At about 14:44, you wrote 4 log base 6 x^4, while since you used the power property, there should be no ^4.

1 answer

Last reply by: Dr Carleen Eaton
Thu Mar 27, 2014 6:56 PM

Post by Christopher Lee on January 22, 2014

At 3:41, shouldn't there be parentheses around x^2-3? Also, there should be a base of 3 when you wrote log y, otherwise it would mean it would have a base of 10.

1 answer

Last reply by: Dr Carleen Eaton
Fri Mar 1, 2013 10:50 PM

Post by Kenneth Montfort on February 26, 2013

I noticed that when you did example IV, you could get two potential factor options. If you subtracted 4x^2 from both sides, your factors end up being (2x + 3)^2, but when you leave the 4x^2 positive on the right, then your factors end up being (2x - 3)^2, why is this?

1 answer

Last reply by: Dr Carleen Eaton
Thu Feb 9, 2012 7:34 PM

Post by Edmund Mercado on February 8, 2012

Dr. Eaton:
At 27:24, did you mean that log16 (18-21)= log16 (-3) not a valid solution?

Properties of Logarithms

  • Use the properties either to convert the log of a complex expression into a combination of logs of simple expressions, or vice versa.
  • To solve a logarithmic equation, first use the properties to combine logs on each side of the equation to get an equation of the form log x = log y. Then equate x and y.

Properties of Logarithms

Simplify
log2( [(x3)/((x + 2)(x − 3)(y − 6)4)] )
  • Recall that there are three properties of logs
  • 1)log(m*n) = log(m) + log(n)
  • 2)log(mn) = nlog(m)
  • 3)log( [m/n] ) = log(m) − log(n)
  • Begin by applying property (3)
  • log2( [(x3)/((x + 2)(x − 3)(y − 6)4)] ) = log2x3 − log2(x + 2)(x − 3)(y − 6)4
  • On the first term, apply property (2)
  • 3log2x − log2(x + 2)(x − 3)(y − 6)4
  • First half is done, now apply property (1) to the second term
  • 3log2x − log2(x + 2)(x − 3)(y − 6)4 = 3log2x − (log2(x + 2) + log2(x − 3) + log2(y − 6)4)
  • Apply property (2) to the very last term
  • = 3log2x − (log2(x + 2) + log2(x − 3) + 4log2(y − 6))
  • Distribute the negative
log2( [(x3)/((x + 2)(x − 3)(y − 6)4)] ) = 3log2x − log2(x + 2) − log2(x − 3) − 4log2(y − 6)
Simplify
log2( [(x3z4)/((x + 2)4(x − 3)2(y − 6)4)] )
  • Recall that there are three properties of logs
  • 1)log(m*n) = log(m) + log(n)
  • 2)log(mn) = nlog(m)
  • 3)log( [m/n] ) = log(m) − log(n)
  • Begin by applying property (3)
  • log2( [(x3z4)/((x + 2)4(x − 3)2(y − 6)4)] ) = log2x3z4 − log2(x + 2)4(x − 3)2(y − 6)4
  • On the first term, apply property (1) followed by property (2)
  • 3log2x + 4log2z − log2(x + 2)4(x − 3)2(y − 6)4
  • First half is done, now apply property (1) to the third term
  • 3log2x + 4log2z − log2(x + 2)4(x − 3)2(y − 6)4 = 3log2x + 4log2z − (log2(x + 2)4 + log2(x − 3)2 + log2(y − 6)4)
  • Apply property (2) to the very last three terms
  • = 3log2x + 4log2z − (4log2(x + 2) + 2log2(x − 3) + 4log2(y − 6))
  • Distribute the negative
log2( [(x3z4)/((x + 2)4(x − 3)2(y − 6)4)] ) = 3log2x + 4log2z − 4log2(x + 2) − 2log2(x − 3) − 4log2(y − 6)
Write as a single logarithm
2log37 + log311 − [1/3]log32
  • Recall that there are three properties of logs
  • 1)log(m*n) = log(m) + log(n)
  • 2)log(mn) = nlog(m)
  • 3)log( [m/n] ) = log(m) − log(n)
  • Condence using rule (2)
  • 2log37 + log311 − [1/3]log32 = log372 + log311 − log32[1/3]
  • Condense uring rule (1)
  • log372 + log311 − log32[1/3] = log3(72*11) − log32[1/3]
  • Condense using rule (3)
  • log3(72*11) − log32[1/3] = log3[((72*11))/(2[1/3])]
  • Simplify
log3[((72*11))/(2[1/3])] = log3[(49*11)/(3√{2})] = log3( [539/(3√{2})] )
Write as a single logarithm
2log3(x − 2) + log3(x + 6) + 3log3(x − 4) − [1/3]log3(y + 2)
  • Recall that there are three properties of logs
  • 1)log(m*n) = log(m) + log(n)
  • 2)log(mn) = nlog(m)
  • 3)log( [m/n] ) = log(m) − log(n)
  • Condence using rule (2)
  • 2log3(x − 2) + log3(x + 6) + 3log3(x − 4) − [1/3]log3(y + 2) = log3(x − 2)2 + log3(x + 6) + log3(x − 4)3 − log3(y + 2)[1/3]
  • Condense uring rule (1)
  • log3(x − 2)2 + log3(x + 6) + log3(x − 4)3 − log3(y + 2)[1/3] = log3(x − 2)2(x + 6)(x − 4)3 − log3(y + 2)[1/3]
  • Condense using rule (3)
  • log3(x − 2)2(x + 6)(x − 4)3 − log3(y + 2)[1/3] = log3[((x − 2)2(x + 6)(x − 4)3)/((y + 2)[1/3])]
  • Simplify
log3[((x − 2)2(x + 6)(x − 4)3)/((y + 2)[1/3])] = log3[((x − 2)2(x + 6)(x − 4)3)/(3√{(y + 2)})]
Solve
log75 + log75x2 = 2
  • Recall that there are three properties of logs
  • 1)log(m*n) = log(m) + log(n)
  • 2)log(mn) = nlog(m)
  • 3)log( [m/n] ) = log(m) − log(n)
  • Condence the left side using property (1)
  • log75 + log75x2 = 2 = > log75*5x2 = 2
  • log725x2 = 2
  • You now have two options: re - write equation in exponential form or find the logyx that equals 2.
  • I'll solving by writing into exponential form
  • 72 = 25x2
  • 49 = 25x2
  • x2 = [49/25]
x = ±√{[49/25]} = ±[7/5]
Solve
log4(x + 5) − log4x = 2
  • Recall that there are three properties of logs
  • 1)log(m*n) = log(m) + log(n)
  • 2)log(mn) = nlog(m)
  • 3)log( [m/n] ) = log(m) − log(n)
  • Condence the left side using property (3)
  • log4(x + 5) − log4x = 2 = > log4( [(x + 5)/x] ) = 2
  • log4( [(x + 5)/x] ) = 2
  • You now have two options: re - write equation in exponential form or find the log4x that equals 2.
  • I'll solve by writing into exponential form
  • 42 = [(x + 5)/x]
  • 16 = [(x + 5)/x]
  • 16x = [(x + 5)/] = 16x = x + 5
  • 15x = 5
  • x = [5/15] = [1/3]
This answer is a valid answer because by pluging into the original equation does not lead to a negative number.
Solve
log2x + log2(x + 36) = log276
  • Recall that there are three properties of logs
  • 1)log(m*n) = log(m) + log(n)
  • 2)log(mn) = nlog(m)
  • 3)log( [m/n] ) = log(m) − log(n)
  • Condence the left side using property (1)
  • log2x + log2(x + 36) = log276 = > log2x*(x + 36) = log276
  • log2x*(x + 36) = log276
  • Since both logs on the left and right side of the equation have the same base, we can continue without the logs.
  • x(x + 36) = 76
  • x2 + 36x = 76
  • x2 + 36x − 76 = 0
  • Factor x2 + 36x − 76 = 0
  • (x + 38)(x − 2) = 0
  • Solve using the Zero Product Property
  • x = − 38
    x = 2
By inspection, you can see that x = − 38 is an erroneous solutions to log2x + log2(x + 36) = log276
since ther will be a negative inside the first term. Since that can't happen, the only solution is x = 2
Solve
log7x − log7(x + 1) = log711
  • Recall that there are three properties of logs
  • 1)log(m*n) = log(m) + log(n)
  • 2)log(mn) = nlog(m)
  • 3)log( [m/n] ) = log(m) − log(n)
  • Condence the left side using property (3)
  • log7x − log7(x + 1) = log711 = > log7[x/((x + 1))] = log711
  • log7[x/((x + 1))] = log711
  • Since both logs on the left and right side of the equation have the same base, we can continue without the logs.
  • [x/(x + 1)] = 11
  • [x/] = 11(x + 1)
  • x = 11x + 1
  • − 1 = 10x
  • x = − [1/10]
  • By inspection, you can see that x = − [1/10] is an erroneous solutions to log7x − log7(x + 1) = log711
  • since ther will be a negative inside the first term.
No Solution
Solve
log92 − log9( − 5x) = log939
  • Recall that there are three properties of logs
  • 1)log(m*n) = log(m) + log(n)
  • 2)log(mn) = nlog(m)
  • 3)log( [m/n] ) = log(m) − log(n)
  • Condence the left side using property (3)
  • log92 − log9( − 5x) = log939 = > log9[2/(( − 5x))] = log939
  • log9[2/(( − 5x))] = log939
  • Since both logs on the left and right side of the equation have the same base, we can continue without the logs.
  • [2/( − 5x)] = 39
  • [2/] = 39( − 5x)
  • 2 = − 195x
  • x = − [2/195]
  • By inspection, you can see that x = − [2/195] is not an erroneous solutions to log92 − log9( − 5x) = log939
  • since there will not be a negative inside the second erm.
x = − [2/195]
Solve
log46 − log4( − 4x) = log467
  • Recall that there are three properties of logs
  • 1)log(m*n) = log(m) + log(n)
  • 2)log(mn) = nlog(m)
  • 3)log( [m/n] ) = log(m) − log(n)
  • Condence the left side using property (3)
  • log46 − log4( − 4x) = log467 = > log4[6/(( − 4x))] = log467
  • log4[6/(( − 4x))] = log467
  • Since both logs on the left and right side of the equation have the same base, we can continue without the logs.
  • [6/( − 4x)] = 67
  • [6/] = 67( − 4x)
  • 6 = − 268x
  • x = − [6/268] = − [3/134]
  • By inspection, you can see that x = − [3/134] is not an erroneous solutions to log46 − log4( − 4x) = log467
  • since there will not be a negative inside the second erm.
x = − [3/134]

*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

Properties of 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
  • Product Property 0:08
    • Example: Product
  • Quotient Property 2:40
    • Example: Quotient
  • Power Property 3:51
    • Moved Exponent
    • Example: Power
  • Equations 5:15
    • Example: Use Properties
  • 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

Transcription: Properties of Logarithms

Welcome to Educator.com.0000

We are going to continue our discussion of logarithms by exploring some properties of logarithms.0002

The first one is the product property: if you have a logb(mn), that is equal to the logb(m) + logb(n).0008

And this might look familiar to you, or this general concept, from working with exponents.0022

Recall that, when you are multiplying two exponents with the same base, you add the exponents.0026

Since logs are a type of exponent, it stands to reason that there would be similarities.0032

So, you are going to see some similarities between the properties we are going to cover today and properties from working with exponents.0040

For example, log9(10) = log9...and I could break this out into factors, 5 times 2...0046

equals the sum of the log of this factor, plus this factor...so the sum of log9(5) and log9(2).0059

Or I could move, instead of from left to right, from right to left.0075

And instead of breaking it apart into 2 separate logs, I may have to combine it into a single log.0079

And combining into a single log is especially helpful when we move on to solving more complex logarithmic equations0084

than the ones we have seen in previous lessons.0091

For example, given log6(5) + log6(7), I may want to combine those.0095

And this would be equivalent to log6...since these have the same base;0105

notice that they have to have the same base for this to work...of 5 times 7, which equals log6(35).0110

This, of course, applies when we are working with variables, as well--with logs involving variables, which is a lot of what we are going to be doing.0120

So, log8(x2 - 36)...I could factor this into log8(x - 6) (x + 6),0128

and then write it as the sum of log8(x - 6) + log8(x + 6).0141

And again, I can move from right to left: if I were given this, I could multiply these two and then combine this into a single logarithm.0149

The quotient property: again, you are going to notice similarities from your work with exponents.0160

If I have logb of the quotient m/n, this is equal to the difference between the log of the dividend and the log of the divisor.0165

Given log2(14/5), I could rewrite this as log2(14) - log2(5).0180

Again, this works when we are talking about logarithms with variables (algebraic logarithms).0197

log3 of something such as (x2 - 3)/y would be equal to log3(x2 - 3) - log3y.0203

So, the log of the dividend minus the log of the divisor--that is the quotient property.0221

The third property we are going to cover is the power property.0232

This one is a bit different that what you have probably seen before.0236

But it states that logb of m to the power p equals p times logb(m).0238

So, if you see what happened: we took this exponent up here and moved it to the front to become a coefficient.0245

Again, you are going to find this property helpful when you are trying to simplify logarithmic equations,0262

or when you need to work with equations to solve them, to be able to move back and forth between these two forms.0270

For example, log4(53): I am going to take this 3 and move it to the front.0278

It is going to become the coefficient: 3log4(5).0290

Or, another example using a variable: log3(x6) =...I pull this out in front...6log3(x).0297

Again, these properties can be used to solve logarithmic equations.0316

Earlier on, we talked about solving logarithmic equations where there was just a log on one side,0320

and situations where we had one log on each side of the equation.0326

Sometimes, however, you are given logarithmic equations where you have multiple logs on each side, sums and differences of logs...0331

And if they have the same base, then you can actually combine those, so that you end up with one log on each side.0339

And then, you can move on to use the techniques previously learned in order to solve those.0345

The product, quotient, and power properties are some extra steps that you might have to take before applying previously-learned techniques.0350

For example, given log7(4x + 50) - 2log7(3) = log7(x + 1) + log7(3),0357

I see that these all have the same bases, so I want to use the property where, if I get a log on one side0373

equal to the log of the other with the same base, then I can just say x = y.0388

But I need to get this into that form; so let's see what we can do to use these properties to combine them.0392

First, I see that I have a coefficient here; so I am going to use the power property.0400

And with the power property, we talked about how logb(mp) = plogb(m).0404

So, I was moving that power to the front to yield this form.0419

I can do the opposite, though: I can take this coefficient and move it up here and make it a power again.0426

And I am going to do that: so, this first log I am just going to keep the same for now: log7(4x + 50).0434

Here, I am going to turn this into log7(3), and I am going to put this back over here and raise the 3 to the second power.0444

That is the only power I have; so I am just going to work with that for now.0458

And I know that 32 is just going to be 9, so I can change that to 9 in this step: it is log7(9).0463

And then, I am going to apply the quotient property on the left, because I have the difference of two logs that have the same base.0482

Using the quotient property, I can rewrite this as the log7(4x + 50) divided by 9.0491

Recall that the quotient property (this was the power property up here) told me that, if I have logb(m/n),0502

that is going to equal logb(m) - logb(n).0519

So here, I am moving from right to left: I am taking these and combining them--that is what I am doing right here.0525

On the right, I am going to use the product property: the product property says that, if I have a logb of a product,0533

that is equal to the sum of the logs of those factors (logb(m) + logb(n)).0543

So, I am going to go ahead and combine these two into a single log, log7(x + 1)(3).0551

Now, I have gotten this in the form that I can actually solve, because I have a log on the left to base 7 and a single log in the right to base 7.0563

And if these are equal, then this expression must equal this expression.0571

Now, I just need to go ahead and solve: 4x + 50, divided by 9, equals...I am going to pull that 3 out in front: 3(x + 1).0575

So, I am going to multiply both sides by 9 to get 4x + 50 = 27(x + 1); 4x + 50 = 27x + 27.0587

I am going to go ahead and subtract a 4x from both sides to get 50 = 23x + 27.0603

Subtract a 27 from both sides: I will get 23 = 23x, so x = 1.0611

Now, I always need to check back and make sure that the solution is valid, because I want to make sure I don't end up taking the log of a negative number.0622

Checking for validity right here: I have log7(4x + 50).0628

Let x equal 1, and check that: log7(4(1) + 50); that is log7(4 + 50), or 54.0636

So, that is OK; let's look right here: the other one I have to check is log7(x + 1), which is log7(1 + 1), and that is just log7(2).0647

So, this one is valid; this is a valid solution.0661

We used this technique, but we had to take a bunch of steps prior to that in order to combine these into a single log on each side of the equation.0667

Before we go on to work some logarithmic equations, let's just talk about simplifying using the properties that we have already covered.0680

And when I am asked to simplify a logarithmic expression, that means that I want to get rid of quotients.0686

I don't want to be taking the log of any quotients, of any products, or of any factors raised to powers.0692

The first thing I am going to do is work with getting rid of this fraction bar.0700

And I am going to use the quotient property for that, because log6(x4y5),0706

divided by this polynomial expression down here, is going to be equal to...let's just say0712

that this whole thing is going to be equal to the log base 6 of the dividend (the numerator), minus log base 6 of the divisor.0723

So, logb(quotient) equals the log of the dividend, minus the log of the divisor.0752

All right, so I got rid of that fraction bar; I don't have any more quotients; but what I do have is a product.0768

I am going to apply the product property to just get rid of this right here.0774

Instead, I am going to say, "OK, I know that this equals the sum of log base 6 of this factor, plus log base 6 of this factor."0781

So, I have to remember that I have my negative sign out here; that is going to apply to each of these terms, once I split them apart.0799

That is log6(x - 3)3 + log6(y + 2)6.0806

All right, I still have one product (I don't have any quotients), so I am going to take care of this one next.0819

This is going to be equal to log6(x4) + log6(y5), minus0827

log6(x - 3)3, plus log6(y + 2)6.0841

Next, I am going to apply the power property.0849

And recall that the power property says that, if I have logb(mp),0853

that is going to be equal to p...this is going to turn into a coefficient...times logb(m).0863

So, I am going to take these powers and move them out front; that is going to give me 4log6(x4), plus0873

5 (that is going to go to the front, also) times log6(y), minus (3 is going to go out in front)0883

3log6(x - 3), plus...the 6 comes out in front...6log6(y + 2).0894

The last thing I want to do is move this negative sign and apply it to each term in here.0906

And that is as far as I can go with simplifying: 4log6...actually, this is gone now,0911

because we have pushed that out in front...plus 5log6y, minus 3log6(x - 3).0918

And then, the negative applies to this term, as well: minus 6log6(y + 2).0929

And this is simplified as far as I can simplify it.0938

I look here, and I no longer have the logs of any quotients; I don't have the logs of any products;0942

and I don't have the logs of any terms or expressions raised to powers.0948

Now, instead of expanding out the expression, I am going to do the opposite.0956

I am asked to write this as a single logarithm: so I am going to instead compress this into one logarithm--the opposite of what I did in the last question.0960

Let's start with the power property: I know that, instead of 3log4(x - 3),0970

I can rewrite this as log4(x - 3), and I am going to turn this into an exponent and write it that way.0975

Plus log4(x + 4): that is going to be raised to the fourth power.0985

Minus log4(x + 7): there is no coefficient there to turn into an exponent.0995

There is here, though: log4(x - 8); and this is going to be raised to the third power.1003

Now, I have the difference here, which means that I can combine this by using the quotient property.1010

I also, within these parentheses, have some addition; so I can use the product property to combine those.1019

And I want to start inside the parentheses; so let's go ahead and do that and use the product property1027

to combine this into log4[(x - 3)3 (x + 4)4].1033

So, it's log base 4 of this times this, minus...here, also, I have a sum; so I can combine that1044

by taking the product of these two factors: log4[(x + 7)(x - 8)3].1056

So, this is this minus this; I can actually leave these brackets off at this point, because I have a single log.1068

Now, we have to apply the quotient property, because we have a difference.1079

This is log4 and log4; and this is going to become the numerator--it is going to become the dividend.1089

(x - 3)3(x + 4)4, divided by (x + 7)(x - 8)3...this is all together.1098

I have written this as a single logarithm, first by applying the power property,1111

then by combining these two logs and these two logs, using the product property,1117

and then finally combining this log and this log, using the quotient property.1122

Now, we are going to apply what we have learned to actually solving logarithmic equations.1129

And we have a technique for solving an equation, as long as we have a single log on each side.1134

I need to combine this side, and I need to also get rid of this 1/2 out in front,1139

so that I can have something of the form logb(x) = logb(y),1145

and then use the property that y must equal x.1151

First, I am going to apply the power property to get rid of these coefficients.1156

log9(23) - log9(2x - 1) = log9(491/2).1161

So, I can do some simplifying: I know that log9(23)...that 2 cubed is 8, so that is log9(8).1176

Now, 491/2 = √49, which equals 7; so I can write this as log9(7)--it is already looking more manageable.1186

On the left, I need to combine these; and I can do that using the quotient property.1199

log9...this is going to go in the numerator, and this will be the denominator.1203

So, this is log9(x) divided by 2x - 1 equals log9(7).1211

Once I am at this point, I have this situation, where I have the same base, and I only have one logarithmic expression on each side.1221

So, I can just say, "OK, x equals y, so 8 divided by 2x - 1 equals 7."1229

I am going to multiply both sides by 2x - 1, which is going to give me 8 = 14x - 7.1236

I am going to add 7 to both sides to get 15 = 14x, and then I am going to divide both sides by 14: so 15/14 = x, or x = 1 and 1/14.1250

It is important that I check the solution; so I want to make sure that I don't end up taking the log of a negative number.1265

And the only log here that has variables as part of it is this one, so I am checking log9(2x - 1), and I am letting x equal 15/14.1271

log9(2) times 15 times 14, minus 1...1284

Now, instead of figuring out this whole thing and subtracting all of that, all I have to do is say,1291

"All right, this is slightly more than 1; so 2 times slightly-more-than-one is going to be slightly more than 2."1297

If I take a value of slightly more than 2 and subtract 1 from it, I am going to be fine; I will have a positive number.1305

I will not be taking the log of a negative number, so this solution is valid.1314

You don't have to get the exact value; you just have to check it far enough to be sure that what you have in here,1320

what you are taking the log of, is not negative; it is greater than 0.1324

OK, Example 4: we are asked to solve a logarithmic equation, and we almost have this form, but not quite.1335

I can't use x = y, because I have this 1/2 here.1346

In order to combine these two, they have to have the same base.1352

In order for us to use the product property to combine these two logs into one, I have to somehow make 1/2 into log16.1355

So, my goal is log16--that I am going to turn 1/2 into that.1366

Well, let's think about this: if I think about my definition of logarithms, I have logb(x) = y.1376

And what I want is log16 of some number (but I don't know what number) is going to equal 1/2,1389

because then, if I have this, instead of writing 1/2 here, I will just write this.1395

Thinking of my definition of logarithms and how I can use that to solve an equation like this,1401

an equation with a log in one side: now I have a separate equation that I need to solve in order to solve this one,1407

just to substitute there: well, recall that logb(x) = y if by = x.1414

So, I am going to solve this by converting it into its exponential form,1424

which is going to give me 161/2 = x.1431

This is the same as √16, which is 4; therefore, log16(4) = 1/2.1438

These are equivalent; since these are equivalent, I can write that up there--I can substitute.1450

Now, my next step is to combine these two logs on the right, using the product property.1469

log16(12x - 21) = log16[4(x2 - 3)].1476

Now, I have it in this form, and I can say, "OK, 12x - 21 = 4(x2 - 3)."1491

And then, I just solve, as usual: this is going to give me 12x - 21 = 4x2 - 12.1500

I have a quadratic equation; I need to first simplify; so let's subtract 12x from both sides.1512

I am going to set the whole equation equal to 0; that is -21 = 4x2 - 12x - 12.1520

I am going to add 21 to both sides to get 0 = 4x2 - 12x + 9.1527

I am going to rewrite this in a more standard form with the variables on the left.1540

Now, this is just a matter of solving the quadratic equation; and this is actually a perfect square: it is (2x - 3) (2x - 3) = 0.1546

And you can check this out to see that 2x times 2x is 4x2; 2x times -3 is -6x, plus -6x, is -12x; -3 times -3 is 9.1558

According to the product property, if 2x - 3 equals 0 (and these are the same, so I only have to look at one factor), then I will have a solution.1576

So, 2x = 3; if I solve for x, x equals 3/2.1585

x = 3/2; now, my last step--this is a potential solution--is that I need to go ahead and check it right here.1590

Let's let x equal 3/2; and we are going to insert that into log16(12x - 21).1599

So, x equals log16(12(3/2) - 21)...not x equals...log16...this cancels;1608

this becomes 1, and this is 6; this is 6 times 3, minus 21; so this is log16(18 - 21).1627

And that gives me log16(-18); therefore, this solution is not valid.1639

And since that is the only solution I have, there is no solution to this equation.1654

The only solution I came up with was an extraneous solution.1664

I solved this by first converting it to this form; and I did that by setting up an equation1668

where I figured out what x had to be for me to write 1/2 in the form log16.1676

And I used the technique of converting this to the exponential form 161/2 = x and solving for x.1682

That told me that log16(4) = 1/2; so instead of writing 1/2 here, I wrote log16(4).1690

Then I used the product property to combine the two logarithmic expressions on the right.1699

Then I used this property, and I set 12x - 21 equal to this expression, and solved, using factoring, a quadratic equation.1705

But then, I went and checked my solution and found that it was not a valid solution.1714

Thanks for visiting Educator.com; that concludes today's lesson.1720

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