MATH 108

Section 4.3

Logarithmic Functions

4

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EXAMPLE 1 Converting from Exponential to Logarithmic Form

Write each exponential equation in logarithmic form.

Solution

6

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EXAMPLE 2 Converting from Logarithmic Form to Exponential Form

Write each logarithmic equation in exponential form.

Solution

3 2

1a log 81 b log

8

(a) 3 raised to what power yields 81?

(b) 2 raised to what power yields ?1

8

9

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EXAMPLE 3 Evaluating Logarithms

Find the value of each of the following logarithms.

Solution

10

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COMMON LOGARITHMS

1. log 10 = 1

2. log 1 = 0

3. log 10x = x

The logarithm with base 10 is called the common logarithm and is denoted

by omitting the base: log x = log10

x. Thus,

y = log x if and only if x = 10 y.

Applying the basic properties of logarithms

11

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NATURAL LOGARITHMS

1. ln e = 1

2. ln 1 = 0

3. log ex = x

The logarithm with base e is called the natural logarithm and is denoted by

ln x. That is, ln x = loge

x. Thus,

y = ln x if and only if x = e y.

Applying the basic properties of logarithms

Natural Logarithm Function

15

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EXAMPLE 7 Using Transformations

Start with the graph of f (x) = log

3

x and use transformations to sketch the

graph of each function.

State the domain and range and the vertical asymptote for the graph of each

function.

2Solve: log 2 1 3 log 343 3xa x b

2(a) Change log 2 1 3 to exponential form.x

32 2 1x 8 2 1x 7

2x

(b) Change log 343 3 to exponential form.x

3 343x 7x

a.

b.

c.

MATH 108

Section 4.4

Properties of Logarithms

20

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RULES OF LOGARITHMSLet M, N, and a be positive real numbers with a ≠ 1, and let r be any real

number.

The logarithm of the product of two (or more) numbers is the sum of the

logarithms of the numbers.

1. Product Rule

21

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RULES OF LOGARITHMSLet M, N, and a be positive real numbers with a ≠ 1, and let r be any real

number.

The logarithm of the quotient of two (or more) numbers is the difference of the

logarithms of the numbers.

2. Quotient Rule

22

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RULES OF LOGARITHMSLet M, N, and a be positive real numbers with a ≠ 1, and let r be any real

number.

The logarithm of a number to the power r is r times the logarithm of the

number.

3. Power Rule

2 32Write log 1 , >1, as a sum of logarithms.

Express all powers as factors.

x x x

4

6 22Write log , 0, as a difference of logarithms.

3

Express all powers as factors.

xx

x

3

2

2Write ln , 2, as a sum and difference of logarithms.

1Express all powers as factors.

x xx

x

27

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EXAMPLE 2 Writing Expressions In Expanded Form

32

2 4

1a. log

2 1

x x

x

Write each expression in expanded form.

3 2 5b. ln x y z

Solution

2

2

Write each of the following as a single logarithm.

a 3ln 2 ln

1b log 4 2log 5

2

c 2log 3 3log 2 log 1

a a

a a a

x

x

29

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EXAMPLE 3 Writing Expressions in Condensed Form

a. log3 log 4x y

Write each expression in condensed form.

21b. 2ln ln 1

2x x

2 2 2c. 2log 5 log 9 log 75

21d. ln ln 1 ln 1

3x x x