Lecture 11 sections 4.3-4.4 logarithmic functions

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


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


EXAMPLE 3 Evaluating Logarithms

Find the value of each of the following logarithms.

Solution

10


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

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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

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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


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



number.

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

logarithms of the numbers.

2. Quotient Rule

22



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


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

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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

Lecture 11 sections 4.3-4.4 logarithmic functions

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Transcript of Lecture 11 sections 4.3-4.4 logarithmic functions