Section 6.4 Logarithmic Functions 183

Chapter 6: Exponential and Logarithmic Functions

Section 6.4: Logarithmic Functions

Exploration 1: Logarithms

Before we define a logarithm, let’s play around with them a little. See if you can follow the

pattern below to be able to fill in the missing pieces to a – f.

1

2

log 3 9 = 2

log 9 3 =

log 4 16 = 2

log 3 27 = 3

(a) log 2 8 = ___

(b) log 4 16 = ___

(c) log ___ 64 = 2

(d) log ___ 64 = 3

(e) log 2 ____ = 4

(f) log 4 2 = ___

Logarithms – A logarithm is just a power

For example, log 2 (32) = 5 says “the logarithm with base 2 of 32 is 5.” It means 2 to the 5th

power is 32. Notice that both in logarithms and exponents, the same number is called the

base.

The logarithmic function with base a, where a > 0 and a ≠ 1 , is denoted by y = log a x

(read as “y is the logarithm to the base a of x”) and is defined by:

________________________________

The domain of the logarithmic function y = logax is ___________.

Example 1*: Convert Exponential to Logarithmic Statements

Change each exponential equation to an equivalent equation involving a logarithm

(a) 58 = t

(b) x −2 = 12

(c) e x = 10

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184

Chapter 6 Exponential and Logarithmic Functions

Example 2*: Convert Logarithmic to Exponential Statements

Change each logarithmic equation to an equivalent equation involving an exponent.

(a) y = log 2 21

(b) log z 12 = 6

(c) log 2 10 = a

Example 3*: Evaluate Logarithmic Expressions

Evaluate the following:

1

(a)* log 3 (81)

(c) log 5 (1)

(b)* log 2

8

(e) log 3 (9)

(f) log 4 (2)

Let’s recall the domain and range

of an exponential function:

Range

Domain

f ( x) = a x

All Real

Numbers

(g) log1/3 (27)

(d) log 2 (16)

(h) log 5 (25)

Since a logarithmic function is the inverse of an

exponential function, fill in the domain and range

below based on what we learned in Section 6.2.

Domain

Range

f ( x) = log a x

All Real

Numbers

greater

than 0

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Section 6.4 Logarithmic Functions 185

Domain and Range of the Logarithmic Function y = log a ( x ) (defining equation x = a y )

Domain:__________________

Range:__________________

Example 4*: Determine the Domain of a Logarithmic Function

Find the domain of each logarithmic function.

x+3

(a) f ( x ) = log3 ( x − 2 )

(b) F ( x ) = log 2

x −1

(c) h ( x ) = log 2 x − 1

(d) g ( x ) = log 1 x 2

2

1.

2.

3.

4.

5.

6.

Properties of the Logarithmic Function f ( x) = log a ( x)

The domain _______________; The range is _______________.

The x-intercept is _______________. There is _______________ y-intercept.

The y-axis ( x = 0 ) is a ____________________ asymptote of the graph.

A logarithmic function is decreasing if __________ and increasing if __________.

The graph of f contains the points ___________________________.

The graph is _______________________________, with no _________________.

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186 Chapter 6 Exponential and Logarithmic Functions

Fact

Natural Logarithm: ln ( x ) means log e ( x ) . It is derived from the Latin phrase,

logarithmus naturalis. In other words, y = ln( x ) if and only if x = e y .

Example 5*: Graph Logarithmic Functions

(a)* Graph f ( x ) = 3ln( x − 1) .

(b)* State the domain of f ( x ) .

(c)* From the graph, determine the range and vertical asymptote of f.

(d) Find f −1 , the inverse of f.

(e) Use f −1 to confirm the range of f found in part (c). From the domain of f, find the range

of f −1 .

(f) Graph f −1 on the same set of axis as f.

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Section 6.4 Logarithmic Functions 187

Fact

Common Logarithm: log ( x ) means log10 ( x ) . In other words, y = log( x ) if and only if

x = 10 y .

Example 6: Graph a Logarithmic Functions

(a) Graph f ( x ) = −2 log ( x + 2 ) .

(b) State the domain of f ( x ) .

(c) From the graph, determine the range and vertical asymptote of f.

(d) Find f −1 , the inverse of f.

(e) Use f −1 to confirm the range of f found in part (c). From the domain of f, find the range

of f −1

(f) Graph f −1 on the same set of axis as f.

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188 Chapter 6 Exponential and Logarithmic Functions

Solving Basic Logarithmic Equations

When solving simple logarithmic equations (they will get more complicated in Section

4.6) follow these steps:

1. Isolate the logarithm if possible.

2. Change the logarithm to exponential form and use the strategies learned in Section

4.3 to solve for the unknown variable.

Example 7*: Solve Logarithmic Equations

Solve the following logarithmic equations

(b)* log x 343 = 3

(a)* log 2 ( 2x +1) = 3

(c) 6 − log

(d) ln ( x ) = 2

(f) log 6 36 = 5 x + 3

(e) 7 log 6 (4 x ) + 5 = −2

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=3

Section 6.4 Logarithmic Functions 189

Steps for solving exponential equations of base e or base 10

1. Isolate the exponential part

2. Change the exponent into a logarithm.

3. Use either the “log” key (if log base 10) or the “ln” (if log base e) key to evaluate the

variable.

Example 8*: Using Logarithms to Solve Exponential Equations

Solve each exponential equation.

(a) e x = 7

(b)* 2e3 x = 6

(c) e5 x −1 = 9

(d) 4(102 x ) + 1 = 21

(e) 3e 2 x +1 − 2 = 10

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190

Chapter 6 Exponential and Logarithmic Functions

Chapter 6: Exponential and Logarithmic Functions

Section 6.5: Properties of Logarithms

Exploration 1: Establish Properties of Logarithms

Calculate the following:

(a) log 5 (1)

(b) log 2 (1)

(c) log(1)

(d) ln(1)

(e) log 5 (5)

(h) ln(e)

(f) log 2 (2)

(g) log(10)

Properties of Logarithms:

To summarize:

1. log a 1 = _______

2. log a a = _______

Exploration 2: Establish Properties of Logarithms

In section 6.4, we found that the inverse of the function f ( x ) = log 2 ( x ) was f −1 ( x ) = 2 x . In

fact, in general we can say that the functions defined by g ( x ) = log a ( x) and h( x) = a x are

inverse functions. Knowing what you know about inverse functions, evaluate:

(a) g (h(r ))

(b) h( g (m))

Properties of Logarithms:

To summarize: In the following properties, M and a are positive real numbers, where a ≠ 1 ,

and r is any real number :

3. log a a r = _______

4. a log a M = _______

Exploration 3: Establish Properties of Logarithms

Show that the following are true

1000

(a) log (100 ⋅ 10 ) = log(100) + log(10) (b) log

= log(1000) − log(100)

100

Copyright © 2016 Pearson Education, Inc.

(c) log103 = 3 log(10)

Section 6.5 Properties of Logarithmic 191

Properties of Logarithms:

To summarize: In the following properties, M, N, and a are positive real numbers, where

a ≠ 1 , and r is any real number :

M

5. log a ( MN ) = __________ 6. log a

N

r

= __________ 7. log a M = ________

Example 1*: Work with the Properties of Logarithms

Use the laws of logarithms to simplify the following:

20

(a) 3log3 18

(b) 2log 2 ( −5)

1

(c) log 1

2 2

(d) ln(e3 )

Example 2: Work with the Properties of Logarithms

Use the laws of logarithms to find the exact value without a calculator.

(b) log 8 (2) − log 8 (32)

(a) log 3 (24) − log 3 (8)

(c) 6log6 (3) + log6 (5)

(d) e

log

e2

(25)

Example 3*: Write a Logarithmic Expression as a Sum or Difference of Logarithms

Write each expression as a as a sum or difference of logarithms. Express all powers as

factors.

x2 y3

2

(a) log 3 ( x − 1)( x + 2 ) , x > 1

(b) log 5

z

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192 Chapter 6 Exponential and Logarithmic Functions

Example 4*: Write a Logarithmic Expression as a Single Logarithm

Write each of the following as a single logarithm.

(b) 3log 6 z − 2 log 6 y

(a) log 2 x + log 2 ( x − 3 )

1

(c) ln ( x − 2 ) + ln x − 5ln ( x + 3)

2

Properties of Logarithms continued:

In the following properties, M, N, and a are positive real numbers where a ≠ 1 :

8. If M = N, then ___________________

9. If log a M = log a N , then ___________

Let a ≠ 1, and b ≠ 1 be positive real numbers. Then the change of base formula says:

10. log a M = _____________

Why would we want to use the change of base formula?

Example 5*: Evaluate a Logarithm Whose Base is Neither 10 nor e.

Approximate the following. Round your answers to four decimal places.

(b) log 7 325

(a) log 3 12

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Section 6.5 Properties of Logarithmic 193

Summary Properties of Logarithms:

In the following properties, M, N, and a are positive real numbers, where a ≠ 1 , and r is any

real number :

log a 1 = _______

log a a = _______

log a M r = _______

a log a M = _______

log a a r = _______

a r = _______

M

log a

N

If M = N, then ___________________

If

log a ( MN ) = ______________

= ______________

log a M = log a N , then ___________

Change of base formula: log a M = _____________

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