Section 3.3 Properties of Logarithms 239

Change of BaseMost calculators have only two types of log keys, one for common logarithms(base 10) and one for natural logarithms (base ). Although common logs andnatural logs are the most frequently used, you may occasionally need to evaluatelogarithms to other bases. To do this, you can use the following change-of-baseformula.

One way to look at the change-of-base formula is that logarithms to base are simply constant multiples of logarithms to base The constant multiplier is

Changing Bases Using Common Logarithms

a.

Use a calculator.

Simplify.

b.

Now try Exercise 1(a).

Changing Bases Using Natural Logarithms

a.

Use a calculator.

Simplify.

b.

Now try Exercise 1(b).

log2 12 �ln 12

ln 2�

2.48491

0.69315� 3.5850

� 2.3219

�3.21888

1.38629

loga x �ln xln a

log4 25 �ln 25

ln 4

log2 12 �log 12

log 2�

1.07918

0.30103� 3.5850

� 2.3219

�1.39794

0.60206

loga x �log x

log a log4 25 �

log 25log 4

1��logba�.b.

a

e

What you should learn• Use the change-of-base

formula to rewrite and evalu-ate logarithmic expressions.

• Use properties of logarithmsto evaluate or rewrite logarith-mic expressions.

• Use properties of logarithmsto expand or condenselogarithmic expressions.

• Use logarithmic functions to model and solve real-lifeproblems.

Why you should learn itLogarithmic functions can beused to model and solve real-lifeproblems. For instance, inExercises 81–83 on page 244, alogarithmic function is used tomodel the relationship betweenthe number of decibels and theintensity of a sound.

Properties of Logarithms

AP Photo/Stephen Chernin

3.3

Change-of-Base FormulaLet and be positive real numbers such that and Then

can be converted to a different base as follows.

Base b Base 10 Base e

loga x �ln xln a

loga x �log xlog a

loga x �logb xlogb a

loga xb � 1.a � 1xb,a,

Example 1

Example 2

333202_0303.qxd 12/7/05 10:29 AM Page 239

Properties of LogarithmsYou know from the preceding section that the logarithmic function with base is the inverse function of the exponential function with base So, it makes sensethat the properties of exponents should have corresponding properties involvinglogarithms. For instance, the exponential property has the correspondinglogarithmic property

For proofs of the properties listed above, see Proofs in Mathematics on page278.

Using Properties of Logarithms

Write each logarithm in terms of ln 2 and ln 3.

a. ln 6 b.

Solutiona. Rewrite 6 as

Product Property

b. Quotient Property

Rewrite 27 as

Power Property

Now try Exercise 17.

Using Properties of Logarithms

Find the exact value of each expression without using a calculator.

a. b.

Solution

a.

b.

Now try Exercise 23.

ln e6 � ln e2 � lne6

e2 � ln e4 � 4 ln e � 4�1� � 4

log5 3�5 � log5 5

1�3 �13 log5 5 �

13 �1� �

13

ln e6 � ln e2log5 3�5

� ln 2 � 3 ln 3

33. � ln 2 � ln 33

ln2

27� ln 2 � ln 27

� ln 2 � ln 3

2 � 3.ln 6 � ln�2 � 3�

ln 2

27

loga1 � 0.a0 � 1

a.a

240 Chapter 3 Exponential and Logarithmic Functions

Encourage your students to know theseproperties well. They will be used forsolving logarithmic and exponentialequations, as well as in calculus.

Remind your students to note thedomain when applying properties oflogarithms to a logarithmic function.For example, the domain of is all real numbers whereas thedomain of is all real numbers x > 0.

g�x� � 2 ln xx � 0,

f �x� � ln x2

There is no general propertythat can be used to rewrite

Specifically,is not equal to

loga u � loga v.loga�u � v�loga�u ± v�.

Historical NoteJohn Napier, a Scottish mathematician, developed logarithms as a way to simplifysome of the tedious calcula-tions of his day. Beginning in1594, Napier worked about 20 years on the invention oflogarithms. Napier was onlypartially successful in his questto simplify tedious calculations.Nonetheless, the developmentof logarithms was a step for-ward and received immediaterecognition.

The

Gra

ng

er C

olle

ctio

n

Properties of LogarithmsLet be a positive number such that and let be a real number. If and are positive real numbers, the following properties are true.

Logarithm with Base a Natural Logarithm

1. Product Property:

2. Quotient Property:

3. Power Property: ln un � n ln uloga un � n loga u

ln u

v� ln u � ln vloga

u

v� loga u � loga v

ln�uv� � ln u � ln vloga�uv� � loga u � loga v

vuna � 1,a

Example 3

Example 4

333202_0303.qxd 12/7/05 3:35 PM Page 240

Rewriting Logarithmic ExpressionsThe properties of logarithms are useful for rewriting logarithmic expressions informs that simplify the operations of algebra. This is true because these proper-ties convert complicated products, quotients, and exponential forms into simplersums, differences, and products, respectively.

Expanding Logarithmic Expressions

Expand each logarithmic expression.

a. b.

Solutiona. Product Property

Power Property

b.

Quotient Property

Power Property

Now try Exercise 47.

In Example 5, the properties of logarithms were used to expand logarithmicexpressions. In Example 6, this procedure is reversed and the properties of loga-rithms are used to condense logarithmic expressions.

Condensing Logarithmic Expressions

Condense each logarithmic expression.

a. b.

c.

Solutiona. Power Property

Product Property

b. Power Property

Quotient Property

c. Product Property

Power Property

Rewrite with a radical.

Now try Exercise 69.

� log2 3�x�x � 1�

� log2�x�x � 1��1�3

13 �log2 x � log2�x � 1�� �13 �log2�x�x � 1��

� ln �x � 2�2

x

2 ln�x � 2� � ln x � ln�x � 2�2 � ln x

� log��x�x � 1�3� 12 log x � 3 log�x � 1� � log x

1�2 � log�x � 1�3

13 �log2 x � log2�x � 1��

2 ln�x � 2� � ln x12 log x � 3 log�x � 1�

�1

2 ln�3x � 5� � ln 7

� ln�3x � 5�1�2 � ln 7

ln �3x � 5

7� ln

�3x � 5�1�2

7

� log4 5 � 3 log4 x � log4 y

log4 5x3y � log4 5 � log4 x3 � log4 y

ln �3x � 5

7log4 5x3y

Section 3.3 Properties of Logarithms 241

A common error made in expandinglogarithmic expressions is to rewrite log as instead of aslog a � n log x.

n log axaxn

A common error made in condensinglogarithmic expressions is to rewrite

as instead of as logxy

.log xlog y

log x � log y

Use a graphing utility to graphthe functions given by

and

in the same viewing window.Does the graphing utility showthe functions with the samedomain? If so, should it?Explain your reasoning.

y2 � ln x

x � 3

y1 � ln x � ln�x � 3�

Exploration

Rewrite using rationalexponent.

Example 5

Example 6

333202_0303.qxd 12/7/05 10:29 AM Page 241

242 Chapter 3 Exponential and Logarithmic FunctionsPe

riod

(in

yea

rs)

Mean distance(in astronomical units)

5

10

15

20

25

30

4 6 8 10x

y

Mercury

Earth

Venus

Mars

Jupiter

Saturn

Planets Near the Sun

2

FIGURE 3.23

ln xVenus

EarthMars

Jupiter

Saturn

31 2

1

2

3

ln y = ln x32

ln y

Mercury

FIGURE 3.24

ApplicationOne method of determining how the - and -values for a set of nonlinear data arerelated is to take the natural logarithm of each of the - and -values. If the pointsare graphed and fall on a line, then you can determine that the - and -values arerelated by the equation

where is the slope of the line.

Finding a Mathematical Model

The table shows the mean distance and the period (the time it takes a planet toorbit the sun) for each of the six planets that are closest to the sun. In the table,the mean distance is given in terms of astronomical units (where Earth’s meandistance is defined as 1.0), and the period is given in years. Find an equation thatrelates and

SolutionThe points in the table above are plotted in Figure 3.23. From this figure it is notclear how to find an equation that relates and To solve this problem, take thenatural logarithm of each of the - and -values in the table. This produces thefollowing results.

Now, by plotting the points in the second table, you can see that all six of thepoints appear to lie in a line (see Figure 3.24). Choose any two points todetermine the slope of the line. Using the two points and you can determine that the slope of the line is

By the point-slope form, the equation of the line is where andYou can therefore conclude that

Now try Exercise 85.

ln y �32 ln x.X � ln x.

Y � ln yY �32 X,

m �0.632 � 00.421 � 0

� 1.5 �32

.

�0, 0�,�0.421, 0.632�

yxx.y

x.y

yx

m

ln y � m ln x

yxyx

yx

Planet Mercury Venus Earth Mars Jupiter Saturn

0.000 0.421 1.649 2.255

0.000 0.632 2.473 3.383�0.486�1.423ln y

�0.324�0.949ln x

Planet Mean Period, ydistance, x

Mercury 0.387 0.241

Venus 0.723 0.615

Earth 1.000 1.000

Mars 1.524 1.881

Jupiter 5.203 11.863

Saturn 9.537 29.447

Example 7

333202_0303.qxd 12/7/05 10:29 AM Page 242

Section 3.3 Properties of Logarithms 243

Exercises 3.3

In Exercises 1–8, rewrite the logarithm as a ratio of (a) com-mon logarithms and (b) natural logarithms.

1. 2.

3. 4.

5. 6.

7. 8.

In Exercises 9–16, evaluate the logarithm using the change-of-base formula. Round your result to threedecimal places.

9. 10.

11. 12.

13. 14.

15. 16.

In Exercises 17–22, use the properties of logarithms torewrite and simplify the logarithmic expression.

17. 18.

19. 20.

21. 22.

In Exercises 23–38, find the exact value of the logarithmicexpression without using a calculator. (If this is not possi-ble, state the reason.)

23. 24.

25. 26.

27. 28.

29. 30.

31.

32.

33.

34.

35.

36.

37.

38.

In Exercises 39–60, use the properties of logarithms toexpand the expression as a sum, difference, and/or constantmultiple of logarithms. (Assume all variables are positive.)

39. 40.

41. 42.

43. 44.

45. 46.

47. 48.

49. 50.

51. 52.

53. 54.

55. 56.

57. 58.

59. 60. ln�x2�x � 2�ln 4�x3�x2 � 3�

log10 xy4

z5log5

x2

y2z3

log2 �x y4

z4ln

x4�y

z5

ln�x2

y3ln 3�x

y

ln 6

�x2 � 1log2

�a � 1

9, a > 1

ln�x2 � 1

x3 �, x > 1ln z�z � 1�2, z > 1

log 4x2 yln xyz2

ln 3�tln �z

log6 1

z3log5

5

x

log10 y

2log8 x

4

log3 10zlog4 5x

log4 2 � log4 32

log5 75 � log5 3

2 ln e6 � ln e5

ln e2 � ln e5

ln 4�e3

ln 1�e

3 ln e4

ln e4.5

log2��16�log3��9�log3 81�0.2log4 161.2

log6 3�6log2

4�8

log5 1

125log3 9

ln 6

e2ln�5e6�

log 9

300log5 1

250

log2�42 � 34�log4 8

log3 0.015log15 1250

log20 0.125log9 0.4

log1�4 5log1�2 4

log7 4log3 7

log7.1 xlog2.6 x

logx 34logx

310

log1�3 xlog1�5 x

log3 xlog5 x

VOCABULARY CHECK:

In Exercises 1 and 2, fill in the blanks.

1. To evaluate a logarithm to any base, you can use the ________ formula.

2. The change-of-base formula for base is given by ________.

In Exercises 3–5, match the property of logarithms with its name.

3. (a) Power Property

4. (b) Quotient Property

5. (c) Product Property

PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com.

loga

uv

� loga u � loga v

ln un � n ln u

loga�uv� � loga u � loga v

loga x �e

333202_0303.qxd 12/7/05 3:36 PM Page 243

In Exercises 61–78, condense the expression to thelogarithm of a single quantity.

61.

62.

63.

64.

65.

66.

67.

68.

69.

70.

71.

72.

73.

74.

75.

76.

77.

78.

In Exercises 79 and 80, compare the logarithmic quantities.If two are equal, explain why.

79.

80.

Sound Intensity In Exercises 81–83, use the followinginformation. The relationship between the number of deci-bels and the intensity of a sound in watts per squaremeter is given by

81. Use the properties of logarithms to write the formula insimpler form, and determine the number of decibels of asound with an intensity of watt per square meter.

82. Find the difference in loudness between an average officewith an intensity of watt per square meter anda broadcast studio with an intensity of wattper square meter.

83. You and your roommate are playing your stereos at thesame time and at the same intensity. How much louder isthe music when both stereos are playing compared withjust one stereo playing?

85. Galloping Speeds of Animals Four-legged animals runwith two different types of motion: trotting and galloping.An animal that is trotting has at least one foot on theground at all times, whereas an animal that is galloping hasall four feet off the ground at some point in its stride. Thenumber of strides per minute at which an animal breaksfrom a trot to a gallop depends on the weight of the animal.Use the table to find a logarithmic equation that relates ananimal’s weight (in pounds) and its lowest gallopingspeed (in strides per minute).y

x

3.16 � 10�51.26 � 10�7

10�6

� � 10 log� I

10�12�.

I�

log7�70, log7 35, 12 � log7 �10

log2 32

log2 4, log2

32

4, log2 32 � log2 4

12�log4�x � 1� � 2 log4�x � 1�� � 6 log4 x

13�log8 y � 2 log8�y � 4�� � log8�y � 1�2�3 ln x � ln�x � 1� � ln�x � 1��

13�2 ln�x � 3� � ln x � ln�x2 � 1��4�ln z � ln�z � 5�� � 2 ln�z � 5�ln x � 4�ln�x � 2� � ln�x � 2��3 log3 x � 4 log3 y � 4 log3 z

log x � 2 log y � 3 log z

2 ln 8 � 5 ln�z � 4�ln x � 3 ln�x � 1��4 log6 2x

14 log3 5x

23 log7�z � 2�2 log2�x � 4�log5 8 � log5 t

log4 z � log4 y

ln y � ln t

ln x � ln 3

244 Chapter 3 Exponential and Logarithmic Functions

84. Human Memory Model Students participating in apsychology experiment attended several lectures andwere given an exam. Every month for a year after theexam, the students were retested to see how much ofthe material they remembered. The average scores forthe group can be modeled by the human memory model

where is the time in months.

(a) Use the properties of logarithms to write the func-tion in another form.

(b) What was the average score on the original exam

(c) What was the average score after 4 months?

(d) What was the average score after 12 months?

(e) Use a graphing utility to graph the function over thespecified domain.

(f) Use the graph in part (e) to determine when theaverage score will decrease to 75.

(g) Verify your answer to part (f) numerically.

�t � 0�?

t

0 ≤ t ≤ 12f �t� � 90 � 15 log�t � 1�,

Model It

Weight, x Galloping Speed, y

25 191.5

35 182.7

50 173.8

75 164.2

500 125.9

1000 114.2

333202_0303.qxd 12/7/05 3:36 PM Page 244

Section 3.3 Properties of Logarithms 245

86. Comparing Models A cup of water at an initial tempera-ture of is placed in a room at a constant temperatureof The temperature of the water is measured every 5minutes during a half-hour period. The results are record-ed as ordered pairs of the form where is the time (inminutes) and is the temperature (in degrees Celsius).

(a) The graph of the model for the data should beasymptotic with the graph of the temperature of theroom. Subtract the room temperature from each of thetemperatures in the ordered pairs. Use a graphingutility to plot the data points and

(b) An exponential model for the data is givenby

Solve for and graph the model. Compare the resultwith the plot of the original data.

(c) Take the natural logarithms of the revised temperatures.Use a graphing utility to plot the points and observe that the points appear to be linear. Use theregression feature of the graphing utility to fit a line tothese data. This resulting line has the form

Use the properties of the logarithms to solve for Verify that the result is equivalent to the model in part (b).

(d) Fit a rational model to the data. Take the reciprocals ofthe -coordinates of the revised data points to generatethe points

Use a graphing utility to graph these points and observethat they appear to be linear. Use the regression featureof a graphing utility to fit a line to these data. Theresulting line has the form

Solve for and use a graphing utility to graph therational function and the original data points.

(e) Write a short paragraph explaining why the transforma-tions of the data were necessary to obtain each model.Why did taking the logarithms of the temperatures leadto a linear scatter plot? Why did taking the reciprocalsof the temperature lead to a linear scatter plot?

Synthesis

True or False? In Exercises 87–92, determine whether thestatement is true or false given that Justify youranswer.

87.

88.

89.

90.

91. If then

92. If then

93. Proof Prove that

94. Proof Prove that

In Exercises 95–100, use the change -of-base formula torewrite the logarithm as a ratio of logarithms. Then use agraphing utility to graph both functions in the sameviewing window to verify that the functions are equivalent.

95. 96.

97. 98.

99. 100.

101. Think About It Consider the functions below.

Which two functions should have identical graphs? Verifyyour answer by sketching the graphs of all three functionson the same set of coordinate axes.

102. Exploration For how many integers between 1 and 20can the natural logarithms be approximated given that

and Approximate these logarithms (do not use a calculator).

Skills Review

In Exercises 103–106, simplify the expression.

103. 104.

105. 106.

In Exercises 107–110, solve the equation.

107. 108.

109. 110.5

x � 1�

2x3

23x � 1

�x4

4x2 � 5x � 1 � 03x2 � 2x � 1 � 0

xy�x�1 � y�1��1�18x3y4��3�18x3y4�3

2x2

3y ��324xy�2

16x�3y

1.6094?ln 5 �ln 2 � 0.6931, ln 3 � 1.0986,

f �x� � ln x

2, g�x� �

ln x

ln 2, h�x� � ln x � ln 2

f �x� � log12.4 xf �x� � log11.8 x

f �x� � log1�4 xf �x� � log1�2 x

f �x� � log4 xf �x� � log2 x

logb un � n logb u.

logb u

v� logb u � logb v.

0 < x < 1.f �x� < 0,

v � u2.f �u� � 2 f �v�,�f �x� �

12 f �x�

x > 2f �x � 2� � f �x� � f �2�,a > 0, x > 0f �ax� � f �a� � f �x�,

f �0� � 0

f �x� � ln x.

T,

1T � 21

� at � b.

t, 1

T � 21�.

y

T.

ln�T � 21� � at � b.

�t, ln�T � 21��

T

T � 21 � 54.4�0.964�t.

�t, T � 21��t, T � 21�.�t, T�

�30, 39.6���25, 42.4��,�20, 46.3��,�15, 51.2��,�10, 57.5��,�5, 66.0��,�0, 78.0��,

Tt�t, T�,

21� C.78� C

333202_0303.qxd 12/7/05 10:29 AM Page 245