(10-20)
Chapter 10
Exponential and Logarithmic Functions
1.58 107 mol/L on July 9, 1998 (Water Resources for
North Carolina, wwwnc.usgs.gov). What was the pH of the
water at that time? 6.8
86. Roanoke River pH. On July 9, 1998 the hydrogen ion concentration of the water in the Roanoke River at Janesville,
North Carolina, was 1.995 107 mol/L (Water Resources for North Carolina, wwwnc.usgs.gov). What was
the pH of the water at that time? 6.7
Roanoke River at Janesville
pH (standard units)
8
t
70
60
50
40
30
20
10
0
6
200 400 600 800 1000 n
Number of infected cows
FIGURE FOR EXERCISE 88
4
GET TING MORE INVOLVED
2
0
a logistic growth model, describes how a disease can spread
very rapidly at first and then very slowly as nearly all of the
population has become infected. 23.5, 27.6, 65.6, 69.1
Time (days)
548
2
3
4
5 6 7
July 1998
8
9
FIGURE FOR EXERCISE 86
Solve each problem.
87. Sound level. The level of sound in decibels (db) is given
by the formula
L 10 log(I 1012),
where I is the intensity of the sound in watts per square
meter. If the intensity of the sound at a rock concert is 0.001
watt per square meter at a distance of 75 meters from the
stage, then what is the level of the sound at this point in the
audience? 90 db
88. Logistic growth. If a rancher has one cow with a contagious disease in a herd of 1,000, then the time in days t for
n of the cows to become infected is modeled by
1000 n
t 5 ln .
999n
Find the number of days that it takes for the disease to
spread to 100, 200, 998, and 999 cows. This model, called
89. Discussion. Use the switch-and-solve method from
Chapter 8 to find the inverse of the function
f(x) 5 log2(x 3). State the domain and range of the
inverse function. f 1(x) 2x5 3, (, ), (3, )
90. Discussion. Find the inverse of the function
f(x) 2 e x4. State the domain and range of the inverse
function.
f 1(x) ln(x 2) 4, (2, ), (, )
G R A P H I N G C ALC U L ATO R
EXERCISES
91. Composition of inverses. Graph the functions y ln(e x)
and y e ln(x). Explain the similarities and differences
between the graphs.
y ln(e x ) x for x , y eln(x) x for
0x
92. The population bomb. The population of the earth is
growing continuously with an annual rate of about 1.6%. If
the present population is 6 billion, then the function
y 6e0.016x gives the population in billions x years from
now. Graph this function for 0 x 200. What will the
population be in 100 years and in 200 years?
29.7 billion, 147.2 billion
10.3 P R O P E R T I E S O F L O G A R I T H M S
In this
section
●
Product Rule for Logarithms
●
Quotient Rule for
Logarithms
●
Power Rule for Logarithms
●
Inverse Properties
●
Using the Properties
The properties of logarithms are very similar to the properties of exponents because
logarithms are exponents. In this section we use the properties of exponents to write
some properties of logarithms. The properties will be used in solving logarithmic
equations in Section 10.4.
Product Rule for Logarithms
If M a x and N ay, we can use the product rule for exponents to write
MN a x a y a xy.
10.3
Properties of Logarithms
(10-21)
549
The equation MN a xy is equivalent to
calculator
loga(MN ) x y.
close-up
Because M a x and N a y are equivalent to x loga(M) and y loga(N ), we
can replace x and y in loga(MN) x y to get
You can illustrate the product
rule for logarithms with a
graphing calculator.
loga(MN ) loga(M) loga(N ).
So the logarithm of a product is the sum of the logarithms, provided that all of the
logarithms are defined. This rule is called the product rule for logarithms.
Product Rule for Logarithms
loga(MN ) loga(M) loga(N)
E X A M P L E
helpful
1
hint
The product rule for logs
comes from the product rule
for exponents. Since you add
exponents when you multiply
exponential expressions (and
logarithms are exponents), it is
not too surprising that the log
of a product is the sum of the
logs.
Using the product rule for logarithms
Write each expression as a single logarithm.
a) log2(7) log2(5)
b) ln(2 ) ln(3 )
Solution
a) log2(7) log2(5) log2(35)
b) ln(2 ) ln(3 ) ln(6 )
Product rule for logarithms
Product rule for logarithms
■
Quotient Rule for Logarithms
If M a x and N a y, we can use the quotient rule for exponents to write
M ax
y a xy.
N
a
M
By the definition of logarithm, N a xy is equivalent to
calculator
close-up
You can illustrate the quotient
rule for logarithms with a
graphing calculator.
M
loga x y.
N
Because x loga(M) and y loga(N ), we have
M
loga loga(M) loga(N ).
N
So the logarithm of a quotient is equal to the difference of the logarithms, provided
that all logarithms are defined. This rule is called the quotient rule for logarithms.
Quotient Rule for Logarithms
M
loga loga(M) loga(N)
N
550
(10-22)
Chapter 10
E X A M P L E
helpful
2
hint
The quotient rule for logs
comes from the quotient rule
for exponents. Since you subtract exponents when you divide exponential expressions,
the log of a quotient is the
difference of the logs.
Exponential and Logarithmic Functions
Using the quotient rule for logarithms
Write each expression as a single logarithm.
a) log2(3) log2(7)
b) ln(w8 ) ln(w2 )
Solution
3
a) log2(3) log2(7) log2 7
8
w
b) ln(w8 ) ln(w2 ) ln 2
w
ln(w6 )
Quotient rule for logarithms
Quotient rule for logarithms
Quotient rule for exponents
■
Power Rule for Logarithms
calculator
If M a x, we can use the power rule for exponents to write
M N (ax )N aNx.
close-up
You can illustrate the power
rule for logarithms with a
graphing calculator.
By the definition of logarithms, MN a Nx is equivalent to
loga(M N ) Nx.
Because x loga(M), we have
loga(M N ) N loga(M).
So the logarithm of a power of a number is the power times the logarithm of the
number, provided that all logarithms are defined. This rule is called the power rule
for logarithms.
Power Rule for Logarithms
loga(M N ) N loga(M )
E X A M P L E
3
Using the power rule for logarithms
Rewrite each logarithm in terms of log(2).
a) log(210 )
helpful
hint
The power rule for logs comes
from the power rule for exponents. Since you multiply
exponents in a power of a
power, the log of a power is
the power times the log.
b) log(2 )
Solution
a) log(210 ) 10 log(2)
b) log(2 ) log(212)
1
log(2)
2
1
c) log log(21)
2
1 log(2)
log(2)
1
c) log 2
Power rule for logarithms
Write 2 as a power of 2.
Power rule for logarithms
1
Write as a power of 2.
2
Power rule for logarithms
■
10.3
Properties of Logarithms
(10-23)
551
Inverse Properties
calculator
close-up
Because logarithmic and exponential functions are inverses, applying one and then
the other returns the original
number.
An exponential function and logarithmic function with the same base are inverses
of each other. For example, the logarithm of 32 base 2 is 5 and the fifth power of 2
is 32. In symbols, we have
2log2(32) 25 32.
If we raise 3 to the fourth power, we get 81; and if we find the base-3 logarithm of
81, we get 4. In symbols, we have
log3(34) log3(81) 4.
We can state the inverse relationship between exponential and logarithm functions
in general with the following inverse properties.
Inverse Properties
1. loga(aM ) M
E X A M P L E
4
2. aloga(M ) M
Using the inverse properties
Simplify each expression.
a) ln(e5 )
b) 2log2(8)
Solution
a) Using the first inverse property, we get ln(e 5) 5.
b) Using the second inverse property, we get 2log2(8) 8.
■
Note that there is more than one way to simplify the expressions in Example 4.
Using the power rule for logarithms and the fact that ln(e) 1, we have
ln(e 5 ) 5 ln(e) 5. Using log2(8) 3, we have 2log2(8) 23 8.
Using the Properties
We have already seen many properties of logarithms. There are three properties that
we have not yet formally stated. Because a1 a and a0 1, we have loga (a) 1
and loga(1) 0 for any positive number a. If we apply the quotient rule to
loga(1N), we get
1
loga loga(1) loga(N) 0 loga(N) loga(N ).
N
So
loga(N). These three new properties along with all of the other
properties of logarithms are summarized as follows.
loga 1
N
Properties of Logarithms
If M, N, and a are positive numbers, a 1, then
2. loga(1) 0
1. loga(a) 1
4. aloga(M ) M
3. loga(aM ) M Inverse properties
5. loga(MN) loga(M) loga(N) Product rule
6. logaM
loga(M) loga(N)
N
7. loga1 loga(N)
N
Quotient rule
8. loga(MN ) N loga(M)
Power rule
552
(10-24)
Chapter 10
Exponential and Logarithmic Functions
We have already seen several ways in which to use the properties of logarithms.
In the next three examples we see more uses of the properties. First we use the rules
of logarithms to write the logarithm of a complicated expression in terms of logarithms of simpler expressions.
E X A M P L E
study
5
tip
Keep track of your time for
one entire week. Account for
how you spend every half
hour. Add up your totals for
sleep, study, work, and recreation. You should be sleeping
50 to 60 hours per week
and studying 1 to 2 hours for
every hour you spend in the
classroom.
calculator
Using the properties of logarithms
Rewrite each expression in terms of log(2) and/or log(3).
a) log(6)
b) log(16)
9
1
d) log c) log 2
3
Solution
a) log(6) log(2 3)
log(2) log(3) Product rule
b) log(16) log(24)
4 log(2) Power rule
9
c) log log(9) log(2)
Quotient rule
2
log(32) log(2)
2 log(3) log(2) Power rule
1
d) log log(3) Property 7
3
close-up
Examine the values of
log(92), log(9) log(2), and
log(9)log(2).
Do not confuse with log9. We can use the quotient rule
CAUTION
to write log9 log(9) log(2),
2
log (9)
log (2)
■
log(9)
log(2)
log (9)
but log (2)
2
log(9) log(2). The expression
means log(9) log(2). Use your calculator to verify these two statements.
The properties of logarithms can be used to combine several logarithms into a
single logarithm (as in Examples 1 and 2) or to write a logarithm of a complicated
expression in terms of logarithms of simpler expressions.
E X A M P L E
6
Using the properties of logarithms
Rewrite each expression as a sum or difference of multiples of logarithms.
(x 3)23
xz
b) log3 a) log y
x
Solution
xz
a) log log(xz) log(y)
y
log(x) log(z) log( y)
Quotient rule
Product rule
(x 3)
b) log3 log3((x 3)23 ) log3(x12 )
x
2
1
log3(x 3) log3(x)
3
2
23
Quotient rule
Power rule
■
10.3
(10-25)
Properties of Logarithms
553
In the next example we use the properties of logarithms to convert expressions
involving several logarithms into a single logarithm. The skills we are learning here
will be used to solve logarithmic equations in Section 10.4.
E X A M P L E
7
Combining logarithms
Rewrite each expression as a single logarithm.
1
1
a) log(x) 2 log(x 1)
b) 3 log( y) log(z) log(x)
2
2
Solution
1
a) log(x) 2 log(x 1) log(x12) log((x 1)2) Power rule
2
x
log 2
Quotient rule
(x 1)
1
b) 3 log( y) log(z) log(x) log( y3) log(z ) log(x) Power rule
2
Product rule
log( y3 z ) log(x)
y3 z
log x
WARM-UPS
Quotient
rule
■
True or false? Explain.
x2
1. log2 log2(x 2) 3 True
8
ln (2)
3. ln(2) True
2
1
1
5. log2 False
8
log2(8)
log(100)
2. log(100) log(10) False
log(10)
7. ln(1) e False
log (100)
8. log(10) False
10
log2(8)
log2(4) False
9. log2(2)
10. 3
4. 3log3(17) 17
True
6. ln(8) 3 ln(2) True
6
10. ln(2) ln(3) ln(7) ln 7
True
EXERCISES
Reading and Writing After reading this section, write out the
answers to these questions. Use complete sentences.
1. What is the product rule for logarithms?
The product rule for logarithms states that
loga(MN) loga(M) loga(N).
2. What is the quotient rule for logarithms?
The quotient rule for logarithms states that
loga(MN) loga(M) loga(N).
3. What is the power rule for logarithms?
The power rule for logarithms states that
loga(MN ) N loga(M).
4. Why is it true that loga(aM ) M?
Since loga(aM ) is the exponent used on a to obtain aM, we
have loga(aM ) M.
5. Why is it true that aloga (M) M?
Since loga(M ) is the exponent you would use on a to obtain
M, using loga(M ) as the exponent produces M :aloga (M) M.
554
(10-26)
Chapter 10
Exponential and Logarithmic Functions
6. Why is it true that loga(1) 0 for a 0 and a 1?
We have loga(1) 0 because a0 1.
Assume all variables involved in logarithms represent numbers
for which the logarithms are defined.
Write each expression as a single logarithm and simplify. See
Example 1.
7. log(3) log(7)
8. ln(5) ln(4)
log(21)
ln(20)
9. log3(5 ) log3(x )
10. ln(x ) ln(y )
ln(xy
log3(5x )
)
2
3
3
11. log(x ) log(x )
12. ln(a ) ln(a5)
log(x 5)
ln(a8)
13. ln(2) ln(3) ln(5) ln(30)
14. log2(x) log2( y) log2(z) log2(xyz)
15. log(x) log(x 3) log(x 2 3x)
16. ln(x 1) ln(x 1) ln(x 2 1)
17. log2(x 3) log2(x 2) log2(x 2 x 6)
18. log3(x 5) log3(x 4) log3(x 2 9x 20)
Write each expression as a single logarithm. See Example 2.
1
19. log(8) log(2) log(4)
20. ln(3) ln(6) ln 2
21. log2(x ) log2(x )
log2(x 4)
23. log(10 ) log(2 )
6
24.
25.
26.
27.
28.
29.
30.
2
22. ln(w ) ln(w )
ln(w6)
log(5 )
9
3
log3(6 ) log3(3
) log3(2 )
ln(4h 8) ln(4) ln(h 2)
log(3x 6) log(3) log(x 2)
log2(w 2 4) log2(w 2) log2(w 2)
log3(k 2 9) log3(k 3) log3(k 3)
ln(x 2 x 6) ln(x 3) ln(x 2)
ln(t 2 t 12) ln(t 4) ln(t 3)
Write each expression in terms of log(3). See Example 3.
1
31. log(27) 3 log(3)
32. log 2 log(3)
9
1
1
4
33. log(3 ) log(3)
34. log(
3 ) log(3)
2
4
x
99
35. log(3 ) x log(3)
36. log(3 ) 99 log(3)
Simplify each expression. See Example 4.
38. ln(e 9 ) 9
39. 5log5(19) 19
37. log2(210) 10
log(2.3)
8
40. 10
2.3
41. log(10 ) 8
42. log4(45) 5
43. e ln(4.3) 4.3
44. 3log3(5.5) 5.5
Rewrite each expression in terms of log(3) and/or log(5). See
Example 5.
45. log(15)
46. log(9)
log(3) log(5)
2 log(3)
5
3
47. log 48. log 3
5
log(5) log(3)
log(3) log(5)
49. log(25)
2 log(5)
51. log(75)
2 log(5) log(3)
1
53. log 3
log(3)
1
50. log 27
3 log(3)
52. log(0.6)
log(3) log(5)
54. log(45)
2 log(3) log(5)
9
56. log 25
2 log(3) 2 log(5)
55. log(0.2)
log(5)
Rewrite each expression as a sum or a difference of multiples of
logarithms. See Example 6.
57. log(xyz) log(x) log(y) log(z)
58. log(3y) log(3) log(y)
60. log2(16y) 4 log2(y)
59. log2(8x) 3 log2(x)
z
x
61. ln ln(x) ln(y)
62. ln ln(z) ln(3)
3
y
63. log(10x 2) 1 2 log(x)
1
64. log(100x ) 2 log(x)
2
(x 3)2
1
65. log5 2 log5(x 3) log5(w)
2
w
(y 6)3
66. log3 3 log3( y 6) log3( y 5)
y5
yzx
67. ln w
ln(y) ln(z) 2 ln(x) ln(w)
(x 1)w
1
68. ln ln(x 1) ln(w) 3 ln(x)
x
2
1
3
Rewrite each expression as a single logarithm. See Example 7.
69. log(x) log(x 1) log(x 2 x)
70. log2(x 2) log2(5) log2(5x 10)
71. ln(3x 6) ln(x 2) ln(3)
72. log3(x 2 1) log3(x 1) log3(x 1)
xz
73. ln(x) ln(w) ln(z) ln w
x
74. ln(x) ln(3) ln(7) ln 21
x 2y3
75. 3 ln(y) 2 ln(x) ln(w) ln w
r 5t3
76. 5 ln(r) 3 ln(t) 4 ln(s) ln s4
1
2
(x 3)12
77. log(x 3) log(x 1) log 2
3
(x 1)23
1
1
78. log( y 4) log(y 4) log(
y2
16 )
2
2
2
1
(x 1)23
79. log2(x 1) log2(x 2) log2 3
4
(x 2)14
10.3
1
80. log3(y 3) 6 log3(y) log3( y 6 y
3 )
2
Determine whether each equation is true or false.
5
log(5)
81. log(56) log(7) log(8)
82. log 9
log(9)
False
False
83. log2(42) (log2(4))2
84. ln(42) (ln(4))2
True
False
85. ln(25) 2 ln(5)
86. ln(3e) 1 ln(3)
True
True
log2(64)
log (16)
87. log2(8)
88. 2 log2(4)
log2(8)
log2(4)
False
True
1
90. log2(8 259) 62
89. log log(3)
3
True
True
5
91. log2(165 ) 20
92. log2 log2(5) 1
2
True
True
93. log(103) 3
94. log3(37) 7
True
True
log (32) 5
95. log(100 3) 2 log(3) 96. 7 log7 (8)
3
False
True
Solve each problem.
97. Growth rate. The annual growth rate for continuous
growth is given by
ln(A) ln(P)
r
,
t
where P is the initial investment and A is the amount
after t years.
a) Rewrite the formula using a single logarithm.
b) In 1998 a share of Microsoft stock was worth 27 times
what it was worth in 1990. What was the annual growth
rate for that period?
a) r ln(A/P)1/t b) 41.2%
Stock value (in dollars)
Growth of Microsoft
120
100
80
60
40
20
0
1990 1992 1994 1996 1998
Year
FIGURE FOR EXERCISE 97
98. Diversity index. The U.S.G.S. measures the quality of a
water sample by using the diversity index d, given by
d [p1 log2( p1) p2 log2( p2) . . .
pn log2( pn)],
(10-27)
Properties of Logarithms
555
where n is the number of different taxons (biological
classifications) represented in the sample and p1 through
pn are the percentages of organisms in each of the n taxons.
The value of d ranges from 0 when all organisms in the
water sample are the same to some positive number when
all organisms in the sample are different. If two-thirds of
the organisms in a water sample are in one taxon and onethird of the organisms are in a second taxon, then n 2
and
2
2
1
1
d log2 log2 .
3
3
3
3
Use the properties of logarithms to write the expression
3
2
on the right-hand side as log2 3
. (In Section 10.4 you
2
will learn how to evaluate a base-2 logarithm using a
calculator.)
GET TING MORE INVOLVED
99. Discussion. Which of the following equations is an
identity? Explain.
a) ln(3x) ln(3) ln(x)
b) ln(3x) ln(3) ln(x)
c) ln(3x) 3 ln(x)
d) ln(3x) ln(x 3)
b
100. Discussion. Which of the following expressions is not
equal to log(523)? Explain.
2
log(5) log(5)
a) log(5)
b) 3
3
1
c) (log(5))23
d) log(25)
3
c
G R A P H I N G C ALC U L ATO R
EXERCISES
101. Graph the functions y1 ln(x ) and y2 0.5 ln(x) on the
same screen. Explain your results.
The graphs are the same because
1
ln(x ) ln(x12) ln(x).
2
102. Graph the functions y1 log(x), y2 log(10x), y3 log(100x), and y4 log(1000x) using the viewing window
2 x 5 and 2 y 5. Why do these curves appear as they do?
Because log(10x) 1 log(x), log(100x) 2 log(x),
and log(1000x) 3 log(x); the graphs lie 1, 2, and 3
units above y log(x).
103. Graph the function y log(e x ). Explain why the graph is
a straight line. What is its slope?
The graph is a straight line because log(ex) x log(e) 0.434x. The slope is log(e) or approximately 0.434.