Outline Logarithm Functions Logarithms Properties

College Algebra & Trigonometry I 4.2 - Logarithm Functions

Math 1100 North Carolina Central University Math & C.S. Department Hicham Qasmi - [email protected]

Math 1100

College Algebra & Trigonometry I

Outline Logarithm Functions Logarithms Properties

1

Logarithm Functions Definition Properties Graphs

2

Logarithms Properties

Math 1100

College Algebra & Trigonometry I

Outline Logarithm Functions Logarithms Properties

1

Logarithm Functions Definition Properties Graphs

2

Logarithms Properties

Math 1100

College Algebra & Trigonometry I

Outline Logarithm Functions Logarithms Properties

Definition Properties Graphs

Outline

1

Logarithm Functions Definition Properties Graphs

2

Logarithms Properties

Math 1100

College Algebra & Trigonometry I

Outline Logarithm Functions Logarithms Properties

Definition Properties Graphs

Logarithm Functions

Definition For x > 0, b > 0 and b 6= 1, we define y = logb x as equivalent to the expression by = x The function f (x) = logb x is the logarithm function with base b.

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College Algebra & Trigonometry I

Outline Logarithm Functions Logarithms Properties

Definition Properties Graphs

Logarithm Functions

Definition For x > 0, b > 0 and b 6= 1, we define y = logb x as equivalent to the expression by = x The function f (x) = logb x is the logarithm function with base b.

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College Algebra & Trigonometry I

Outline Logarithm Functions Logarithms Properties

Definition Properties Graphs

Logarithm Functions Example 1

Write each equation in the equivalent exponential form: 2 = log2 x 3 = logb 64 log3 7 = y

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College Algebra & Trigonometry I

Outline Logarithm Functions Logarithms Properties

Definition Properties Graphs

Logarithm Functions Example 2

Write each equation in the equivalent logarithm form: 1

122 = x

2

b3 = 8

3

ey = 9

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College Algebra & Trigonometry I

Outline Logarithm Functions Logarithms Properties

Definition Properties Graphs

Logarithm Functions Example 3

Evaluate 1

log2 16

2

log3 9

3

log2 55

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College Algebra & Trigonometry I

Outline Logarithm Functions Logarithms Properties

Definition Properties Graphs

Outline

1

Logarithm Functions Definition Properties Graphs

2

Logarithms Properties

Math 1100

College Algebra & Trigonometry I

Outline Logarithm Functions Logarithms Properties

Definition Properties Graphs

Basic Properties

Theorem 1

logb b = 1

2

logb 1 = 0

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College Algebra & Trigonometry I

Outline Logarithm Functions Logarithms Properties

Definition Properties Graphs

Basic Properties

Theorem 1

logb b = 1

2

logb 1 = 0

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College Algebra & Trigonometry I

Outline Logarithm Functions Logarithms Properties

Definition Properties Graphs

Example

Evaluate 1

log7 7

2

log5 1

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College Algebra & Trigonometry I

Outline Logarithm Functions Logarithms Properties

Definition Properties Graphs

Inverse Properties

Theorem 1

logb bx = 1

2

logb 1 = 0

Math 1100

College Algebra & Trigonometry I

Outline Logarithm Functions Logarithms Properties

Definition Properties Graphs

Inverse Properties

Theorem 1

logb bx = 1

2

logb 1 = 0

Math 1100

College Algebra & Trigonometry I

Outline Logarithm Functions Logarithms Properties

Definition Properties Graphs

Inverse Properties

Theorem 1

logb bx = 1

2

logb 1 = 0

Math 1100

College Algebra & Trigonometry I

Outline Logarithm Functions Logarithms Properties

Definition Properties Graphs

Inverse Properties

Theorem 1

logb bx = 1

2

logb 1 = 0

Math 1100

College Algebra & Trigonometry I

Outline Logarithm Functions Logarithms Properties

Definition Properties Graphs

Example

Evaluate 1

log4 45

2

6log6 9

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College Algebra & Trigonometry I

Outline Logarithm Functions Logarithms Properties

Definition Properties Graphs

Outline

1

Logarithm Functions Definition Properties Graphs

2

Logarithms Properties

Math 1100

College Algebra & Trigonometry I

Outline Logarithm Functions Logarithms Properties

Definition Properties Graphs

Example of Graph Graph f (x) = 2x and g(x) = log2 x in the same coordinate system. 5 4

y = 2x

3 2 1 0 −5 −4 −3 −2 −1 0 −1

1

2

3

4

5

−2 −3 −4 −5 Math 1100

College Algebra & Trigonometry I

Outline Logarithm Functions Logarithms Properties

Definition Properties Graphs

Example of Graph Graph f (x) = 2x and g(x) = log2 x in the same coordinate system. 5 4

y = 2x

reflection

3 2 1 0 −5 −4 −3 −2 −1 0 −1

1

2

3

4

5

−2 −3 −4 −5 Math 1100

College Algebra & Trigonometry I

Outline Logarithm Functions Logarithms Properties

Definition Properties Graphs

Example of Graph Graph f (x) = 2x and g(x) = log2 x in the same coordinate system. 5 4

y = 2x

reflection

3 2 1 0 −5 −4 −3 −2 −1 0 −1

y = log2 x 1

2

3

4

5

−2 −3 −4 −5 Math 1100

College Algebra & Trigonometry I

Outline Logarithm Functions Logarithms Properties

Definition Properties Graphs

Characteristics of the Graph of f (x) = logb x

1

The x-intercept is 1.

2

There is no y-intercept.

3

The y-axis (or x = 0) is a vertical asymptote.

4

If b > 1, the function is increasing. If 0 < b < 1, the function is decreasing.

5

The graph is smooth and continuous.

6

The domain is (0, ∞)

Math 1100

College Algebra & Trigonometry I

Outline Logarithm Functions Logarithms Properties

Definition Properties Graphs

Characteristics of the Graph of f (x) = logb x

1

The x-intercept is 1.

2

There is no y-intercept.

3

The y-axis (or x = 0) is a vertical asymptote.

4

If b > 1, the function is increasing. If 0 < b < 1, the function is decreasing.

5

The graph is smooth and continuous.

6

The domain is (0, ∞)

Math 1100

College Algebra & Trigonometry I

Outline Logarithm Functions Logarithms Properties

Definition Properties Graphs

Characteristics of the Graph of f (x) = logb x

1

The x-intercept is 1.

2

There is no y-intercept.

3

The y-axis (or x = 0) is a vertical asymptote.

4

If b > 1, the function is increasing. If 0 < b < 1, the function is decreasing.

5

The graph is smooth and continuous.

6

The domain is (0, ∞)

Math 1100

College Algebra & Trigonometry I

Outline Logarithm Functions Logarithms Properties

Definition Properties Graphs

Characteristics of the Graph of f (x) = logb x

1

The x-intercept is 1.

2

There is no y-intercept.

3

The y-axis (or x = 0) is a vertical asymptote.

4

If b > 1, the function is increasing. If 0 < b < 1, the function is decreasing.

5

The graph is smooth and continuous.

6

The domain is (0, ∞)

Math 1100

College Algebra & Trigonometry I

Outline Logarithm Functions Logarithms Properties

Definition Properties Graphs

Characteristics of the Graph of f (x) = logb x

1

The x-intercept is 1.

2

There is no y-intercept.

3

The y-axis (or x = 0) is a vertical asymptote.

4

If b > 1, the function is increasing. If 0 < b < 1, the function is decreasing.

5

The graph is smooth and continuous.

6

The domain is (0, ∞)

Math 1100

College Algebra & Trigonometry I

Outline Logarithm Functions Logarithms Properties

Definition Properties Graphs

Characteristics of the Graph of f (x) = logb x

1

The x-intercept is 1.

2

There is no y-intercept.

3

The y-axis (or x = 0) is a vertical asymptote.

4

If b > 1, the function is increasing. If 0 < b < 1, the function is decreasing.

5

The graph is smooth and continuous.

6

The domain is (0, ∞)

Math 1100

College Algebra & Trigonometry I

Outline Logarithm Functions Logarithms Properties

Definition Properties Graphs

Characteristics of the Graph of f (x) = logb x

1

The x-intercept is 1.

2

There is no y-intercept.

3

The y-axis (or x = 0) is a vertical asymptote.

4

If b > 1, the function is increasing. If 0 < b < 1, the function is decreasing.

5

The graph is smooth and continuous.

6

The domain is (0, ∞)

Math 1100

College Algebra & Trigonometry I

Outline Logarithm Functions Logarithms Properties

Definition Properties Graphs

Domain example

Example Find the domain of f (x) = log4 (x + 3)

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Outline Logarithm Functions Logarithms Properties

Definition Properties Graphs

Common Logarithm

The common logarithm is the logarithm with base b = 10: log x = log10 x

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Outline Logarithm Functions Logarithms Properties

Definition Properties Graphs

Natural Logarithm

The naturel logarithm is the logarithm with base b = e: ln x = loge x

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Outline Logarithm Functions Logarithms Properties

Logarithms Properties

Theorem Let b, M, N and p be positive real numbers with b 6= 1. Then, 1

2

3

logb (MN) = logb M + logb N M logb ( ) = logb M − logb N N logb (M p ) = p · logb M

Math 1100

College Algebra & Trigonometry I

Outline Logarithm Functions Logarithms Properties

Logarithms Properties

Theorem Let b, M, N and p be positive real numbers with b 6= 1. Then, 1

2

3

logb (MN) = logb M + logb N M logb ( ) = logb M − logb N N logb (M p ) = p · logb M

Math 1100

College Algebra & Trigonometry I

Outline Logarithm Functions Logarithms Properties

Logarithms Properties

Theorem Let b, M, N and p be positive real numbers with b 6= 1. Then, 1

2

3

logb (MN) = logb M + logb N M logb ( ) = logb M − logb N N logb (M p ) = p · logb M

Math 1100

College Algebra & Trigonometry I