seia2e_1103 chapter 11. exponential and logarithmic functions

8/2/2019 Seia2e_1103 Chapter 11. Exponential and Logarithmic Functions

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Chapter 11

Exponentialand

LogarithmicFunctions

Section 3

LogarithmicFunctions


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Sullivan, III & Struve,Elementary and Intermediate Algebra 11.3 - 2Copyright 2010 Pearson Education, Inc.

Section 11.3 Objectives

1 Change Exponential Expressions to LogarithmicExpressions

2 Change Logarithmic Expressions to ExponentialExpressions

3 Evaluate Logarithmic Functions

4 Determine the Domain of a Logarithmic Function

5 Graph Logarithmic Functions

6 Work with Natural and Common Logarithms

7 Solve Logarithmic Equations

8 Study Logarithmic Models That Describe Our

World


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Logarithms

y = logax x = ay

The logarithmic function to the base a, where a > 0 and a 0,is denoted byy = logax (read as y is the logarithm to the base a

ofx) and is defined by

y = loga

x is equivalent to x = ay

Exponent =y

Base = a


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xponen a qua ons n o ogForm

Exponential Form Logarithmic Form

50 = 1 log51 = 0

23 = 8 log28 = 3

( )4

1 1=2 16 1 2

1log = 416

2 16 =36

6

1log = 236

Example:Write the following exponential equations in logarithmic form.


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Log Equations into ExponentialForm

Exponential FormLogarithmic Form

101 = 10log1010= 1

25

= 32log232= 5

( )2

1 1=3 91 3

1log = 29

14 =64

3

41log = 364

Example:Write the following logarithmic equations in exponential form.


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Solving Logarithmic Equations

Example:Solve the following logarithmic equation:

Convert into an equivalent exponential equation.

a) logx 81 = 4 b) log10x = 2

a) logx 81 = 4

x4 = 81

Solve the exponential equation.x4 = 34

x = 3

Convert into an equivalent exponential equation.

b) log10x = 2

10 2 =x

Simplify.

1

100

=x

E l i L i h i


7/17Sullivan, III & Struve,Elementary and Intermediate Algebra 11.3 - 7Copyright 2010 Pearson Education, Inc.

Evaluating LogarithmicFunctions

Example:

Find the exact value of log381.

Lety = log381.

Write the logarithm as an exponent.

To find the exact value of a logarithm, we write the logarithm inexponential notation and use the fact that ifau = av, u = v.

y = log381

Set the exponents equal to each other.

3y = 81

81 = 343y = 34

y = 4

Therefore, log381= 4.

E l i L i h i



Evaluating LogarithmicFunctions

Example:Find the valuef(125) given thatf(x) = log5x.

Lety = log5125.

Write the logarithm as an exponent.

y = log5125

Set the exponents equal to each other.

5y = 125

125 = 535y = 53

y = 3

Therefore,f(125) = 3



Domain and Range

Domain and Range of the Logarithmic FunctionDomain of the logarithmic function = (0, )

Range of the logarithmic function = (, )

Example:Find the domain off(x) = log3(x 2).

The domain is the set of all real numbers such thatx 2 > 0.

x 2 > 0x > 2

The domain offis {x|x > 2} or (2, ).

G L t



Grap ng a Logar t m cFunction

Example:Graphy = log2x.

y

x

222

2

24

12

01

1

2

yx

1412

The exponential form of

the function isx = 2y

y = log2x

y = 2x

y = log2x is the inverse function ofy = 2x.



roper es o ogar m cFunctions

Properties of the Graph of a Logarithmic

Function f(x) = logax, a > 1

1. The domain is the set of all positive real numbers.

The range is the set of all real numbers.

2. There is noy-intercept; thex-intercept is 1.

3. The graph offcontains the points(1,0), and ( ,1).a

1

a,1

,



roper es o ogar m cFunctions

Properties of the Graph of a Logarithmic

Function f(x) = logax, 0 < a < 1

1. The domain is the set of all positive real numbers.

The range is the set of all real numbers.

2. There is noy-intercept; thex-intercept is 1.

3. The graph offcontains the points(1,0), and ( ,1).a

1

a,1

,

T N t C



T e Natura an CommonLogarithm

If the base on a logarithmic function is the numbere, we havethe naturallogarithm function. This function is denoted by the

symbol, ln.

The natural logarithm:y = lnx if and only if x = e

y

.

If the base on a logarithmic function is the number 10, we have

the commonlogarithm function. If the base a of the

logarithmic function is not indicated, it is understood to be 10.

The common logarithm:y = logx if and only if x = 10y.

T N t C



2.98318

Press this key on

the calculator.

Example:

Using a calculator, evaluate each of the following:

T e Natura an CommonLogarithm

a) log 962

962 log

Some calculators require the

button to be entered

first.

log

b) log 73.2

1.8645173.2 log




Example:Solve the following logarithmic equation: log(2x + 3) = 1

Write as an exponential equation.

log(2x + 3) = 1

101 = 2x + 3

Subtract 3 from both sides.7 = 2x

Divide each side by 2.72

x =

Check:log 2 7

2+ 3

= 1

log 7 + 3( )= 1

log10 1=



pp ca on o ogar m:Loudness

The loudnessL, measured in decibels, of a sound of

intensityx, measured in watts per square meter, is

L x( )= 10log x1012


17/17Sullivan III & Struve Elementary and Intermediate Algebra 11 3 17Copyright 2010 Pearson Education Inc


Example:Normal conversation has an intensity level of 1012 watt per

square meter. How many decibels is normal conversation?

The loudness of normal conversation is 60 decibels.

L 10

6( )= 10log 106

1012

= 10log106 12( )

= 10log106

= 10 6( )= 60 decibels

seia2e_1103 chapter 11. exponential and logarithmic functions

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