Properties of Logarithms 2 Discovery b x ln b c p x fx b x...

Graphing Exponential and Log Functions

Study TipYou can obtain a partial table of coordinates for without having to obtain and reverse coordinates for Because means we begin withvalues for and compute corresponding values for

–2 –1

1

0

2

1

4

2

8

3

x

g(x) ! log2x

14

12Start with

values for g (x ).

14

Use x = 2g (x ) to compute x . For example,if g (x ) � 2, x � 2 2 � �� .1

22

x:g1x2 2g1x2 = x,g1x2 = log2 xf1x2 = 2x.g1x2 = log2 x

Graphs of Exponential and Logarithmic Functions

Graph and in the same rectangular coordinate system.

Solution We first set up a table of coordinates for Reversing thesecoordinates gives the coordinates for the inverse function

We now plot the ordered pairs from each table, connecting them with smoothcurves. Figure 3.7 shows the graphs of and its inverse function

The graph of the inverse can also be drawn by reflecting the graph ofabout the line y = x.f1x2 = 2x

g1x2 = log2 x.f1x2 = 2x

–2 –1 0

1

1

2

2

4

3

8

x

Reversecoordinates.

f(x) ! 2x 14

12 –2 –1

1

0

2

1

4

2

8

3

x

g(x) ! log2x

14

12

g1x2 = log2 x.f1x2 = 2x.

g1x2 = log2 xf1x2 = 2x

EXAMPLE 6

1

1234

78

65

2 3

1 2 3 4 8765 1 2 3

y

x

g (x ) � log2 x

f(x ) � 2x

y � x

Figure 3.7 The graphs ofand its inverse functionf1x2 = 2x

y

x

y

x

(0, 1)

(1, 0)

f 1(x ) � logb xf 1(x ) � logb x

f (x ) � bx

f (x ) � bx

(0, 1)

(1, 0)

b � 1 0 � b � 1

y � x y � x

Horizontal asymptote:y � 0

Horizontal asymptote:y � 0

Vertical asymptote:x � 0


Figure 3.8 Graphs ofexponential and logarithmicfunctions

Check Point 6 Graph and in the same rectangularcoordinate system.

Figure 3.8 illustrates the relationship between the graph of an exponentialfunction, shown in blue, and its inverse, a logarithmic function, shown in red, forbases greater than 1 and for bases between 0 and 1. Also shown and labeled are theexponential function’s horizontal asymptote and the logarithmic function’svertical asymptote 1x = 02. 1y = 02

g1x2 = log3 xf1x2 = 3x

404 Chapter 3 Exponential and Logarithmic Functions

P-BLTZMC03_387-458-hr 19-11-2008 11:42 Page 404

The Domain of a Logarithmic FunctionIn Section 3.1, we learned that the domain of an exponential function of the form

includes all real numbers and its range is the set of positive real numbers.Because the logarithmic function reverses the domain and the range of theexponential function, the domain of a logarithmic function of the formis the set of all positive real numbers. Thus, is defined because the value of in the logarithmic expression, 8, is greater than zero and therefore is included in thedomain of the logarithmic function However, and are not defined because 0 and are not positive real numbers and therefore areexcluded from the domain of the logarithmic function In general,the domain of consists of all for which

Finding the Domain of a Logarithmic Function

Find the domain of

Solution The domain of consists of all for which Solving thisinequality for we obtain Thus, the domain of is This isillustrated in Figure 3.13. The vertical asymptote is and all points on thegraph of have that are greater than

Check Point 7 Find the domain of

Common LogarithmsThe logarithmic function with base 10 is called the common logarithmic function.The function is usually expressed as A calculator witha key can be used to evaluate common logarithms. Here are some examples:! LOG !

f1x2 = log x.f1x2 = log10 x

f1x2 = log41x - 52.-3.x-coordinatesfx = -3

1-3, q2.fx 7 -3.x,x + 3 7 0.xf

f1x2 = log41x + 32.EXAMPLE 7

g(x)>0.xf(x) ! logb g(x)f1x2 = log2 x.

-8log21-82log2 0f1x2 = log2 x.

xlog2 8f(x) ! logb x

f1x2 = bx

Here are some other examples of transformations of graphs of logarithmicfunctions:

• The graph of is the graph of shifted up threeunits, shown in Figure 3.10.

• The graph of is the graph of reflected about theshown in Figure 3.11.

• The graph of is the graph of reflected aboutthe shown in Figure 3.12.y-axis,

f1x2 = log2 xr1x2 = log21-x2x-axis,f1x2 = log2 xh1x2 = - log2 x

f1x2 = log4 xg1x2 = 3 + log4 x

3

2

1

1

2

3

4

5

1 2 76543

y

x

g (x ) � 3 ��log4 x

f(x ) � log4 x


Figure 3.10 Shifting vertically upthree units

3

2

1

1

2

3

1 2 76543x

y

h(x ) � log2 x

f(x ) � log2 x


Figure 3.11 Reflection about the x-axis

3

2

1

1

2

3

1 2 543 1 2 5 4 3

y

x

r(x ) � log2 ( x ) f(x ) � log2 x


Figure 3.12 Reflection about the y-axis

3

2

1

1

2

3

1 2 3 1 2

y

x

x � 3

f(x ) � log4 (x � 3)

Figure 3.13 The domain ofis 1-3, q2.f1x2 = log41x + 32

! Use common logarithms.

" Find the domain of a logarithmicfunction.


P-BLTZMC03_387-458-hr 19-11-2008 11:42 Page 406

The Domain of a Logarithmic FunctionIn Section 3.1, we learned that the domain of an exponential function of the form

includes all real numbers and its range is the set of positive real numbers.Because the logarithmic function reverses the domain and the range of theexponential function, the domain of a logarithmic function of the formis the set of all positive real numbers. Thus, is defined because the value of in the logarithmic expression, 8, is greater than zero and therefore is included in thedomain of the logarithmic function However, and are not defined because 0 and are not positive real numbers and therefore areexcluded from the domain of the logarithmic function In general,the domain of consists of all for which

Finding the Domain of a Logarithmic Function

Find the domain of

Solution The domain of consists of all for which Solving thisinequality for we obtain Thus, the domain of is This isillustrated in Figure 3.13. The vertical asymptote is and all points on thegraph of have that are greater than

Check Point 7 Find the domain of

Common LogarithmsThe logarithmic function with base 10 is called the common logarithmic function.The function is usually expressed as A calculator witha key can be used to evaluate common logarithms. Here are some examples:! LOG !

f1x2 = log x.f1x2 = log10 x

f1x2 = log41x - 52.-3.x-coordinatesfx = -3

1-3, q2.fx 7 -3.x,x + 3 7 0.xf

f1x2 = log41x + 32.EXAMPLE 7

g(x)>0.xf(x) ! logb g(x)f1x2 = log2 x.

-8log21-82log2 0f1x2 = log2 x.

xlog2 8f(x) ! logb x

f1x2 = bx

Here are some other examples of transformations of graphs of logarithmicfunctions:

• The graph of is the graph of shifted up threeunits, shown in Figure 3.10.

• The graph of is the graph of reflected about theshown in Figure 3.11.

• The graph of is the graph of reflected aboutthe shown in Figure 3.12.y-axis,

f1x2 = log2 xr1x2 = log21-x2x-axis,f1x2 = log2 xh1x2 = - log2 x

f1x2 = log4 xg1x2 = 3 + log4 x

3

2

1

1

2

3

4

5

1 2 76543

y

x

g (x ) � 3 ��log4 x

f(x ) � log4 x


Figure 3.10 Shifting vertically upthree units

3

2

1

1

2

3

1 2 76543x

y

h(x ) � log2 x

f(x ) � log2 x


Figure 3.11 Reflection about the x-axis

3

2

1

1

2

3

1 2 543 1 2 5 4 3

y

x

r(x ) � log2 ( x ) f(x ) � log2 x


Figure 3.12 Reflection about the y-axis

3

2

1

1

2

3

1 2 3 1 2

y

x

x � 3

f(x ) � log4 (x � 3)

Figure 3.13 The domain ofis 1-3, q2.f1x2 = log41x + 32

! Use common logarithms.

" Find the domain of a logarithmicfunction.


P-BLTZMC03_387-458-hr 19-11-2008 11:42 Page 406

Section 3.2 Logarithmic Functions 405

The red graphs in Figure 3.8 illustrate the following general characteristics oflogarithmic functions:

Table 3.4 Transformations Involving Logarithmic FunctionsIn each case, represents a positive real number.c

Transformation Equation Description

Vertical translation

g1x2 = logb x - c

g1x2 = logb x + c • Shifts the graph of upward units.

• Shifts the graph of downward units.c

f1x2 = logb xc

f1x2 = logb x

Horizontal translation

g1x2 = logb1x - c2g1x2 = logb1x + c2 • Shifts the graph of

to the left units.Vertical asymptote:

• Shifts the graph of to the right units.Vertical asymptote: x = c

cf1x2 = logb xx = -c

cf1x2 = logb x

Reflection

g1x2 = logb1-x2g1x2 = - logb x • Reflects the graph of about the

• Reflects the graph of about the y-axis.

f1x2 = logb xx-axis.

f1x2 = logb x

Vertical stretchingor shrinking

g1x2 = c logb x • Vertically stretches the graph ofif

• Vertically shrinks the graph ofif 0 6 c 6 1.f1x2 = logb x

c 7 1.f1x2 = logb x

Horizontal stretchingor shrinking

g1x2 = logb1cx2 • Horizontally shrinks the graph ofif

• Horizontally stretches the graph ofif 0 6 c 6 1.f1x2 = logb x

c 7 1.f1x2 = logb x

Characteristics of Logarithmic Functions of the Form

1. The domain of consists of all positive real numbers:The range of consists of all real numbers:

2. The graphs of all logarithmic functions of the form passthrough the point (1, 0) because The is 1.There is no

3. If has a graph that goes up to the right and is anincreasing function.

4. If has a graph that goes down to the right and is adecreasing function.

5. The graph of approaches, but does not touch, the The or is a vertical asymptote. As x : 0+, logb x : - q or q .x = 0,y-axis,

y-axis.f1x2 = logb x

0 6 b 6 1, f1x2 = logb x

b 7 1, f1x2 = logb xy-intercept.

x-interceptf112 = logb 1 = 0.f1x2 = logb x1- q , q2.f1x2 = logb x

10, q2.f1x2 = logb x

f(x) ! logb x

The graphs of logarithmic functions can be translated vertically orhorizontally, reflected, stretched, or shrunk. These transformations aresummarized in Table 3.4.

For example, Figure 3.9 illustrates that the graph of is thegraph of shifted one unit to the right. If a logarithmic function istranslated to the left or to the right, both the and the vertical asymptoteare shifted by the amount of the horizontal shift. In Figure 3.9, the of is1. Because is shifted one unit to the right, its is 2. Also observe that thevertical asymptote for the or is shifted one unit to the right for thevertical asymptote for Thus, is the vertical asymptote for g.x = 1g.

x = 0,y-axis,f,x-interceptg

fx-interceptx-intercept

f1x2 = log2 xg1x2 = log21x - 12

3

2

1

1

2

3

1 2 76543

y

x

x � 1

g (x ) � log2 (x 1)

f(x ) � log2 x


Figure 3.9 Shifting one unit to the right

f1x2 = log2 x

P-BLTZMC03_387-458-hr 19-11-2008 11:42 Page 405

Section 3.2 Logarithmic Functions 405

The red graphs in Figure 3.8 illustrate the following general characteristics oflogarithmic functions:

Table 3.4 Transformations Involving Logarithmic FunctionsIn each case, represents a positive real number.c

Transformation Equation Description

Vertical translation

g1x2 = logb x - c

g1x2 = logb x + c • Shifts the graph of upward units.

• Shifts the graph of downward units.c

f1x2 = logb xc

f1x2 = logb x

Horizontal translation

g1x2 = logb1x - c2g1x2 = logb1x + c2 • Shifts the graph of

to the left units.Vertical asymptote:

• Shifts the graph of to the right units.Vertical asymptote: x = c

cf1x2 = logb xx = -c

cf1x2 = logb x

Reflection

g1x2 = logb1-x2g1x2 = - logb x • Reflects the graph of about the

• Reflects the graph of about the y-axis.

f1x2 = logb xx-axis.

f1x2 = logb x

Vertical stretchingor shrinking

g1x2 = c logb x • Vertically stretches the graph ofif

• Vertically shrinks the graph ofif 0 6 c 6 1.f1x2 = logb x

c 7 1.f1x2 = logb x

Horizontal stretchingor shrinking

g1x2 = logb1cx2 • Horizontally shrinks the graph ofif

• Horizontally stretches the graph ofif 0 6 c 6 1.f1x2 = logb x

c 7 1.f1x2 = logb x

Characteristics of Logarithmic Functions of the Form

1. The domain of consists of all positive real numbers:The range of consists of all real numbers:

2. The graphs of all logarithmic functions of the form passthrough the point (1, 0) because The is 1.There is no

3. If has a graph that goes up to the right and is anincreasing function.

4. If has a graph that goes down to the right and is adecreasing function.

5. The graph of approaches, but does not touch, the The or is a vertical asymptote. As x : 0+, logb x : - q or q .x = 0,y-axis,

y-axis.f1x2 = logb x

0 6 b 6 1, f1x2 = logb x

b 7 1, f1x2 = logb xy-intercept.

x-interceptf112 = logb 1 = 0.f1x2 = logb x1- q , q2.f1x2 = logb x

10, q2.f1x2 = logb x

f(x) ! logb x

The graphs of logarithmic functions can be translated vertically orhorizontally, reflected, stretched, or shrunk. These transformations aresummarized in Table 3.4.

For example, Figure 3.9 illustrates that the graph of is thegraph of shifted one unit to the right. If a logarithmic function istranslated to the left or to the right, both the and the vertical asymptoteare shifted by the amount of the horizontal shift. In Figure 3.9, the of is1. Because is shifted one unit to the right, its is 2. Also observe that thevertical asymptote for the or is shifted one unit to the right for thevertical asymptote for Thus, is the vertical asymptote for g.x = 1g.

x = 0,y-axis,f,x-interceptg

fx-interceptx-intercept

f1x2 = log2 xg1x2 = log21x - 12

3

2

1

1

2

3

1 2 76543

y

x

x � 1

g (x ) � log2 (x 1)

f(x ) � log2 x


Figure 3.9 Shifting one unit to the right

f1x2 = log2 x

P-BLTZMC03_387-458-hr 19-11-2008 11:42 Page 405

Properties of Logarithms

Section 3.3 Properties of Logarithms 413

134. Graph each of the following functions in the same viewingrectangle and then place the functions in order from the onethat increases most slowly to the one that increases mostrapidly.

Critical Thinking ExercisesMake Sense? In Exercises 135–138, determine whethereach statement makes sense or does not make sense, and explainyour reasoning.

135. I’ve noticed that exponential functions and logarithmicfunctions exhibit inverse, or opposite, behavior in many ways.For example, a vertical translation shifts an exponentialfunction’s horizontal asymptote and a horizontal translationshifts a logarithmic function’s vertical asymptote.

136. I estimate that lies between 1 and 2 because and

137. I can evaluate some common logarithms without having touse a calculator.

138. An earthquake of magnitude 8 on the Richter scale is twiceas intense as an earthquake of magnitude 4.

In Exercises 139–142, determine whether each statement is true orfalse. If the statement is false, make the necessary change(s) toproduce a true statement.

139.

140.

141. The domain of is

142. is the exponent to which must be raised to obtain

143. Without using a calculator, find the exact value of

log3 81 - logp 1log212 8 - log 0.001

.

x.blogb x

1- q , q2.f1x2 = log2 x

log1-1002 = -2

log2 8log2 4

= 84

82 = 64.81 = 8log8 16

y = x, y = 1x , y = ex, y = ln x, y = xx, y = x2

144. Without using a calculator, find the exact value of

145. Without using a calculator, determine which is the greaternumber: or

Group Exercise146. This group exercise involves exploring the way we grow.

Group members should create a graph for the function thatmodels the percentage of adult height attained by a boy whois years old, Let

find function values, and connect the resultingpoints with a smooth curve. Then create a graph for thefunction that models the percentage of adult height attainedby a girl who is years old, Let

find function values, and connect theresulting points with a smooth curve. Group membersshould then discuss similarities and differences in the growthpatterns for boys and girls based on the graphs.

Preview ExercisesExercises 147–149 will help you prepare for the material covered inthe next section. In each exercise, evaluate the indicated logarithmicexpressions without using a calculator.

147. a. Evaluate:b. Evaluate:c. What can you conclude about or

148. a. Evaluate:b. Evaluate:c. What can you conclude about

149. a. Evaluate:b. Evaluate:c. What can you conclude about

log3 81, or log3 9 2?

2 log3 9.log3 81.

log2 16, or log2a322b?

log2 32 - log2 2.log2 16.

log218 # 42?log2 32,log2 8 + log2 4.log2 32.

x = 5, 6, 7, Á , 15,g1x2 = 62 + 35 log1x - 42.x

3, Á , 12,x = 1, 2, f1x2 = 29 + 48.8 log1x + 12.x

log3 40.log4 60

log43log31log2 824.

Objectives

! Use the product rule.

" Use the quotient rule.

# Use the power rule.

$ Expand logarithmicexpressions.

% Condense logarithmicexpressions.

& Use the change-of-baseproperty.

Properties of Logarithms

We all learn new things in different ways. Inthis section, we consider important proper-

ties of logarithms. What would be the mosteffective way for you to learn these properties?Would it be helpful to use your graphing utilityand discover one of these properties for your-self? To do so, work Exercise 133 in ExerciseSet 3.2 before continuing. Would it be helpful

to evaluate certain logarithmic expressions thatsuggest three of the properties? If this is the

case, work Preview Exercises 147–149 in ExerciseSet 3.2 before continuing. Would the propertiesbecome more meaningful if you could see exactlywhere they come from? If so, you will find details

of the proofs of many of these properties inAppendix A.The remainder of our work in this

chapter will be based on the properties oflogarithms that you learn in this section.

Sec t i on 3.3

P-BLTZMC03_387-458-hr 19-11-2008 11:42 Page 413

DiscoveryWe know that Showthat you get the same result by writing100,000 as and then usingthe product rule. Then verify theproduct rule by using other numberswhose logarithms are easy to find.

1000 # 100

log 100,000 = 5.


The Product RuleLet and be positive real numbers with

The logarithm of a product is the sum of the logarithms.

logb1MN2 = logb M + logb N

b Z 1.Nb, M,

When we use the product rule to write a single logarithm as the sum of twologarithms, we say that we are expanding a logarithmic expression. For example, wecan use the product rule to expand

Using the Product Rule

Use the product rule to expand each logarithmic expression:

a. b.

Solution

a. The logarithm of a product is the sum of thelogarithms.

b. The logarithm of a product is the sum of thelogarithms. These are common logarithms withbase 10 understood.

Because then log 10 = 1.logb b = 1, = 1 + log x

log110x2 = log 10 + log x

log417 # 52 = log4 7 + log4 5

log110x2.log417 # 52EXAMPLE 1

ln (7x) = ln 7 + ln x.

The logarithmof a product

the sum ofthe logarithms.

is

ln17x2:


The Product RuleProperties of exponents correspond to properties of logarithms. For example, whenwe multiply with the same base, we add exponents:

This property of exponents, coupled with an awareness that a logarithm is anexponent, suggests the following property, called the product rule:

bm # bn = bm + n.

Check Point 1 Use the product rule to expand each logarithmic expression:

a. b.

The Quotient RuleWhen we divide with the same base, we subtract exponents:

This property suggests the following property of logarithms, called the quotient rule:

bm

bn = bm - n.

log1100x2.log617 # 112

The Quotient RuleLet and be positive real numbers with

The logarithm of a quotient is the difference of the logarithms.

logbaMNb = logb M - logb N

b Z 1.Nb, M,

DiscoveryWe know that Show thatyou get the same result by writing 16

as and then using the quotient rule.

Then verify the quotient rule usingother numbers whose logarithms areeasy to find.

322

log2 16 = 4.


P-BLTZMC03_387-458-hr 19-11-2008 11:42 Page 414


When we use the quotient rule to write a single logarithm as the difference oftwo logarithms, we say that we are expanding a logarithmic expression. For example,

we can use the quotient rule to expand

Using the Quotient Rule

Use the quotient rule to expand each logarithmic expression:

a. b.

Solution

a. The logarithm of a quotient is the difference of thelogarithms.

b.The logarithm of a quotient is the difference of thelogarithms. These are natural logarithms withbase understood.

Because then

Check Point 2 Use the quotient rule to expand each logarithmic expression:

a. b.

The Power RuleWhen an exponential expression is raised to a power, we multiply exponents:

This property suggests the following property of logarithms, called the power rule:

1bm2n = bmn.

ln¢ e5

11≤ .log8a 23

xb

ln e3 = 3.ln ex = x, = 3 - ln 7

e ln¢ e3

7≤ = ln e3 - ln 7

log7a 19xb = log7 19 - log7 x

ln¢ e3

7≤ .log7a 19

xb

EXAMPLE 2

log a b = log x - log 2.

The logarithmof a quotient

the difference ofthe logarithms.

is

x2

log x2

:

The Power RuleLet and be positive real numbers with and let be any real number.

The logarithm of a number with an exponent is the product of the exponent andthe logarithm of that number.

logb Mp = p logb M

pb Z 1,Mb

! Use the power rule.

When we use the power rule to “pull the exponent to the front,” we say that weare expanding a logarithmic expression. For example, we can use the power rule toexpand

ln x2 = 2 ln x.

The logarithm ofa number with an

exponent

the product of theexponent and the

logarithm of that number.

is

ln x2:

P-BLTZMC03_387-458-hr 19-11-2008 11:42 Page 415

Domain: ( !, 0) ´ (0, !) Domain: (0, !)

y � 2 ln xy � ln x2

Figure 3.18 and have different domains.2 ln xln x2

When expanding a logarithmic expression, you might want to determinewhether the rewriting has changed the domain of the expression. For the rest of thissection, assume that all variables and variable expressions represent positive numbers.

Using the Power Rule

Use the power rule to expand each logarithmic expression:

a. b. c.

Solution

a. The logarithm of a number with an exponent is the exponenttimes the logarithm of the number.

b. Rewrite the radical using a rational exponent.

Use the power rule to bring the exponent to the front.

c. We immediately apply the power rule because the entire variable expression, is raised to the 5th power.

Check Point 3 Use the power rule to expand each logarithmic expression:

a. b. c.

Expanding Logarithmic ExpressionsIt is sometimes necessary to use more than one property of logarithms when youexpand a logarithmic expression. Properties for expanding logarithmic expressionsare as follows:

log1x + 422.ln13 xlog6 39

4x,log14x25 = 5 log14x2 = 1

2 ln x

ln1x = ln x

12

log5 74 = 4 log5 7

log14x25.ln1xlog5 74

EXAMPLE 3


Figure 3.18 shows the graphs of and in byviewing rectangles. Are and the same? The graphs illustrate thatand have different domains. The graphs are only the same if

Thus, we should write

ln x2 = 2 ln x for x 7 0.

x 7 0.y = 2 ln xy = ln x2

2 ln xln x23-5, 5, 14 3-5, 5, 14y = 2 ln xy = ln x2

Properties for Expanding Logarithmic ExpressionsFor and

1. Product rule

2. Quotient rule

3. Power rulelogb Mp = p logb M



N 7 0:M 7 0

! Expand logarithmic expressions.

P-BLTZMC03_387-458-hr 19-11-2008 11:42 Page 416


Try to avoid the following errors:

Incorrect!

logb1MNp2 = p logb1MN2 logb Mlogb N

= logb M - logb N

logbaMNb =

logb Mlogb N

logb1M # N2 = logb M # logb N logb1M - N2 = logb M - logb N logb1M + N2 = logb M + logb N

Study TipThe graphs show that

In general,

y2 � ln x � ln 3

y1 � ln (x � 3)

[ 4, 5, 1] by [ 3, 3, 1]

logb1M + N2 Z logb M + logb N.

ln (x+3) ! ln x+ln 3.

y � ln x shifted3 units left

y � ln x shiftedln 3 units up

Expanding Logarithmic Expressions

Use logarithmic properties to expand each expression as much as possible:

a. b.

Solution We will have to use two or more of the properties for expandinglogarithms in each part of this example.

a. Use exponential notation.

Use the product rule.

Use the power rule.

b. Use exponential notation.

Use the quotient rule.

Use the product rule on

Use the power rule.

Apply the distributive property.

because 2 is the power to which we must raise 6 to get 36.

Check Point 4 Use logarithmic properties to expand each expression as muchas possible:

a. b. log5¢ 1x

25y3 ≤ .logb Ax413 y B162 = 362log6 36 = 2 = 1

3 log6 x - 2 - 4 log6 y

= 13

log6 x - log6 36 - 4 log6 y

= 13

log6 x - 1log6 36 + 4 log6 y2 log6136y42. = log6 x

13

- 1log6 36 + log6 y42 = log6 x

13

- log6 136 y42 log6 ¢ 13 x

36 y4 ≤ = log6 ¢ x

13

36 y4 ≤ = 2 logb x + 12

logb y

= logb x2 + logb y

12

logb1x21y2 = logbAx2

y

12

Blog6 ¢ 13 x

36 y4 ≤ .logb1x21y2EXAMPLE 4

P-BLTZMC03_387-458-hr 19-11-2008 11:42 Page 417

3.3


1000 # 100

log 100,000 = 5.





b Z 1.Nb, M,




a. b.

Solution





log417 # 52 = log4 7 + log4 5


ln (7x) = ln 7 + ln x.



is

ln17x2:




bm # bn = bm + n.


a. b.



bm

bn = bm - n.

log1100x2.log617 # 112




b Z 1.Nb, M,




322

log2 16 = 4.


P-BLTZMC03_387-458-hr 19-11-2008 11:42 Page 414


1000 # 100

log 100,000 = 5.





b Z 1.Nb, M,




a. b.

Solution





log417 # 52 = log4 7 + log4 5


ln (7x) = ln 7 + ln x.



is

ln17x2:




bm # bn = bm + n.


a. b.



bm

bn = bm - n.

log1100x2.log617 # 112




b Z 1.Nb, M,




322

log2 16 = 4.


P-BLTZMC03_387-458-hr 19-11-2008 11:42 Page 414


When we use the quotient rule to write a single logarithm as the difference oftwo logarithms, we say that we are expanding a logarithmic expression. For example,

we can use the quotient rule to expand

Using the Quotient Rule

Use the quotient rule to expand each logarithmic expression:

a. b.

Solution

a. The logarithm of a quotient is the difference of thelogarithms.

b.The logarithm of a quotient is the difference of thelogarithms. These are natural logarithms withbase understood.

Because then

Check Point 2 Use the quotient rule to expand each logarithmic expression:

a. b.

The Power RuleWhen an exponential expression is raised to a power, we multiply exponents:

This property suggests the following property of logarithms, called the power rule:

1bm2n = bmn.

ln¢ e5

11≤ .log8a23

xb

ln e3 = 3.ln ex = x, = 3 - ln 7

e ln¢ e3

7≤ = ln e3 - ln 7

log7a19xb = log7 19 - log7 x

ln¢ e3

7≤ .log7a19

xb

EXAMPLE 2

log a b = log x - log 2.

The logarithmof a quotient

the difference ofthe logarithms.

is

x2

log x2

:

The Power RuleLet and be positive real numbers with and let be any real number.

The logarithm of a number with an exponent is the product of the exponent andthe logarithm of that number.

logb Mp = p logb M

pb Z 1,Mb

! Use the power rule.

When we use the power rule to “pull the exponent to the front,” we say that weare expanding a logarithmic expression. For example, we can use the power rule toexpand

ln x2 = 2 ln x.

The logarithm ofa number with an

exponent

the product of theexponent and the

logarithm of that number.

is

ln x2:

P-BLTZMC03_387-458-hr 19-11-2008 11:42 Page 415

Domain: ( !, 0) ´ (0, !) Domain: (0, !)

y � 2 ln xy � ln x2

Figure 3.18 and have different domains.2 ln xln x2

When expanding a logarithmic expression, you might want to determinewhether the rewriting has changed the domain of the expression. For the rest of thissection, assume that all variables and variable expressions represent positive numbers.

Using the Power Rule

Use the power rule to expand each logarithmic expression:

a. b. c.

Solution

a. The logarithm of a number with an exponent is the exponenttimes the logarithm of the number.

b. Rewrite the radical using a rational exponent.

Use the power rule to bring the exponent to the front.

c. We immediately apply the power rule because the entire variable expression, is raised to the 5th power.

Check Point 3 Use the power rule to expand each logarithmic expression:

a. b. c.

Expanding Logarithmic ExpressionsIt is sometimes necessary to use more than one property of logarithms when youexpand a logarithmic expression. Properties for expanding logarithmic expressionsare as follows:

log1x + 422.ln13 xlog6 39

4x,log14x25 = 5 log14x2 = 1

2 ln x

ln1x = ln x

12

log5 74 = 4 log5 7

log14x25.ln1xlog5 74

EXAMPLE 3


Figure 3.18 shows the graphs of and in byviewing rectangles. Are and the same? The graphs illustrate thatand have different domains. The graphs are only the same if

Thus, we should write

ln x2 = 2 ln x for x 7 0.

x 7 0.y = 2 ln xy = ln x2

2 ln xln x23-5, 5, 14 3-5, 5, 14y = 2 ln xy = ln x2

Properties for Expanding Logarithmic ExpressionsFor and

1. Product rule

2. Quotient rule

3. Power rulelogb Mp = p logb M



N 7 0:M 7 0

! Expand logarithmic expressions.

P-BLTZMC03_387-458-hr 19-11-2008 11:42 Page 416

4. 3 2 4( ) log ( )f x x � �

Domain:

Range:

x-intercept:

Asymptote:

Increasing Interval:

Decreasing Interval:

End Behavior:

5. 5( ) log( )f x x � �

Domain:

Range:

x-intercept:

Asymptote:



End Behavior:

6. 13

1 2( ) log ( )f x x � �

Domain:

Range:

x-intercept:

Asymptote:



End Behavior:

© Gina Wilson (All Things Algebra®, LLC), 2017

4. 3 2 4( ) log ( )f x x � �

Domain:

Range:

x-intercept:

Asymptote:



End Behavior:

5. 5( ) log( )f x x � �

Domain:

Range:

x-intercept:

Asymptote:



End Behavior:

6. 13

1 2( ) log ( )f x x � �

Domain:

Range:

x-intercept:

Asymptote:



End Behavior:


Name: _______________________________ Unit 4: Exponential & Logarithmic Functions

Date: _____________________ Per: _______

Homework 6: Graphing Logarithmic Functions

Directions: Graph each function, then identify its key characteristics. 1. 2( ) logf x x Domain:

Range:

x-intercept:

Asymptote:



End Behavior:

2. 14

2( ) logf x x � Domain:

Range:

x-intercept:

Asymptote:



End Behavior:

3. 5( ) ln( )f x x � � Domain:

Range:

x-intercept:

Asymptote:



End Behavior:

** This is a 2-page document! **


2.

3.

4.𝑓(𝑥) = −ln(𝑥 + 5)

3.𝑓(𝑥) = log12(𝑥 + 1)

− 2

2.𝑓(𝑥) = log2(𝑥 − 2) + 4

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