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7.4
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495
a x and loga x
We have defined general exponential functions such as 2x, 10x , and px . In this section we
compute their derivatives and integrals. We also define the general logarithmic functions
such as log2 x, log10 x, and logp x, and find their derivatives and integrals as well.
The Derivative of au
We start with the definition a x = e x ln a :
d x
d x ln a
d
a =
e
= e x ln a #
sx ln ad
dx
dx
dx
d u
du
e = eu
dx
dx
= a x ln a.
If a 7 0, then
d x
a = a x ln a.
dx
With the Chain Rule, we get a more general form.
If a 7 0 and u is a differentiable function of x, then a u is a differentiable function
of x and
d u
du
a = a u ln a
.
dx
dx
(1)
These equations show why e x is the exponential function preferred in calculus. If a = e,
then ln a = 1 and the derivative of a x simplifies to
d x
e = e x ln e = e x .
dx
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Chapter 7: Transcendental Functions
EXAMPLE 1
x
y  1  y y 10 x
10 
x
y 3x
y 2x
y 1
3
x
y 1
2
y
1
–1
0
1
1x
x
FIGURE 7.12 Exponential functions
decrease if 0 6 a 6 1 and increase if
a 7 1 . As x : q , we have a x : 0 if
0 6 a 6 1 and a x : q if a 7 1 . As
x : - q , we have a x : q if 0 6 a 6 1
and a x : 0 if a 7 1 .
Differentiating General Exponential Functions
(a)
d x
3 = 3x ln 3
dx
(b)
d -x
d
3 = 3-x sln 3d s -xd = -3-x ln 3
dx
dx
(c)
d sin x
d
3
= 3sin x sln 3d ssin xd = 3sin x sln 3d cos x
dx
dx
From Equation (1), we see that the derivative of a x is positive if ln a 7 0, or a 7 1,
and negative if ln a 6 0, or 0 6 a 6 1. Thus, a x is an increasing function of x if a 7 1
and a decreasing function of x if 0 6 a 6 1. In each case, a x is one-to-one. The second
derivative
d2 x
d x
sa d =
sa ln ad = sln ad2 a x
dx
dx 2
is positive for all x, so the graph of a x is concave up on every interval of the real line
(Figure 7.12).
Other Power Functions
The ability to raise positive numbers to arbitrary real powers makes it possible to define
functions like x x and x ln x for x 7 0. We find the derivatives of such functions by rewriting
the functions as powers of e.
EXAMPLE 2
Differentiating a General Power Function
Find dy>dx if y = x x,
Solution
x 7 0.
Write x x as a power of e:
y = x x = e x ln x .
a x with a = x.
Then differentiate as usual:
dy
d x ln x
e
=
dx
dx
= e x ln x
d
sx ln xd
dx
1
= x x ax # x + ln xb
= x x s1 + ln xd.
The Integral of au
If a Z 1, so that ln a Z 0, we can divide both sides of Equation (1) by ln a to obtain
au
du
1 d u
sa d.
=
dx
ln a dx
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Integrating with respect to x then gives
L
au
du
d u
1 d u
1
1 u
dx =
sa d dx =
sa d dx =
a + C.
dx
ln
a
dx
ln
a
dx
ln
a
L
L
Writing the first integral in differential form gives
L
EXAMPLE 3
(a)
L
(b)
L
a u du =
au
+ C.
ln a
(2)
Integrating General Exponential Functions
2x dx =
2x
+ C
ln 2
Eq. (2) with a = 2, u = x
2sin x cos x dx
=
2u du =
L
2sin x
=
+ C
ln 2
2u
+ C
ln 2
u = sin x, du = cos x dx, and Eq. (2)
u replaced by sin x
Logarithms with Base a
As we saw earlier, if a is any positive number other than 1, the function a x is one-to-one
and has a nonzero derivative at every point. It therefore has a differentiable inverse. We
call the inverse the logarithm of x with base a and denote it by log a x.
y
DEFINITION
loga x
For any positive number a Z 1,
y 2x
loga x is the inverse function of a x .
yx
y log 2 x
2
1
0
The graph of y = loga x can be obtained by reflecting the graph of y = a x across the 45°
line y = x (Figure 7.13). When a = e, we have log e x = inverse of e x = ln x. Since log a x
and a x are inverses of one another, composing them in either order gives the identity function.
x
1 2
Inverse Equations for a x and log a x
a loga x = x
FIGURE 7.13 The graph of 2x and its
inverse, log 2 x .
loga sa x d = x
sx 7 0d
(3)
sall xd
(4)
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Chapter 7: Transcendental Functions
EXAMPLE 4
5
Applying the Inverse Equations
(a) log2 s2 d = 5
(c) 2log2 s3d = 3
(b) log10 s10 -7 d = -7
(d) 10 log10 s4d = 4
Evaluation of loga x
The evaluation of loga x is simplified by the observation that loga x is a numerical multiple
of ln x.
loga x =
ln x
1 #
ln x =
ln a
ln a
(5)
We can derive this equation from Equation (3):
a loga sxd = x
ln a loga sxd = ln x
loga sxd # ln a = ln x
loga x =
ln x
ln a
Eq. (3)
Take the natural logarithm of both sides.
The Power Rule in Theorem 2
Solve for loga x.
For example,
log10 2 =
0.69315
ln 2
L 0.30103
L
2.30259
ln 10
The arithmetic rules satisfied by log a x are the same as the ones for ln x (Theorem 2).
These rules, given in Table 7.2, can be proved by dividing the corresponding rules for the
natural logarithm function by ln a. For example,
TABLE 7.2
Rules for base a
logarithms
For any numbers x 7 0 and
y 7 0,
1.
2.
Product Rule:
loga xy = loga x + loga y
Quotient Rule:
x
loga y = loga x - loga y
3.
Reciprocal Rule:
ln xy = ln x + ln y
ln xy
ln y
ln x
=
+
ln a
ln a
ln a
loga xy = loga x + loga y.
Rule 1 for natural logarithms Á
Á divided by ln a Á
Á gives Rule 1 for base a logarithms.
Derivatives and Integrals Involving loga x
To find derivatives or integrals involving base a logarithms, we convert them to natural
logarithms.
If u is a positive differentiable function of x, then
d ln u
d
1 d
1 # 1 du
sloga ud =
a
b =
sln ud =
.
dx
dx ln a
ln a dx
ln a u dx
1
loga y = -loga y
4.
Power Rule:
loga x y = y loga x
d
1 # 1 du
sloga ud =
dx
ln a u dx
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7.4
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499
EXAMPLE 5
3
d
1
s3x + 1d =
3x + 1 dx
sln 10ds3x + 1d
(a)
d
1
log10 s3x + 1d =
dx
ln 10
(b)
log2 x
ln x
1
x dx = ln 2
x dx
L
L
#
log2 x =
ln x
ln 2
=
1
u du
ln 2 L
=
2
sln xd2
1 u2
1 sln xd
+ C
+ C =
+ C =
ln 2 2
ln 2 2
2 ln 2
u = ln x,
1
du = x dx
Base 10 Logarithms
Base 10 logarithms, often called common logarithms, appear in many scientific formulas. For example, earthquake intensity is often reported on the logarithmic Richter scale.
Here the formula is
a
Magnitude R = log10 a b + B,
T
where a is the amplitude of the ground motion in microns at the receiving station, T is the period of the seismic wave in seconds, and B is an empirical factor that accounts for the weakening of the seismic wave with increasing distance from the epicenter of the earthquake.
EXAMPLE 6
Earthquake Intensity
For an earthquake 10,000 km from the receiving station, B = 6.8. If the recorded vertical
ground motion is a = 10 microns and the period is T = 1 sec, the earthquake’s magnitude
is
R = log10 a
10
b + 6.8 = 1 + 6.8 = 7.8.
1
An earthquake of this magnitude can do great damage near its epicenter.
The pH scale for measuring the acidity of a solution is a base 10 logarithmic scale.
The pH value (hydrogen potential) of the solution is the common logarithm of the reciprocal of the solution’s hydronium ion concentration, [H3 O+]:
Most foods are acidic spH 6 7d.
Food
Bananas
Grapefruit
Oranges
Limes
Milk
Soft drinks
Spinach
pH Value
4.5–4.7
3.0–3.3
3.0–4.0
1.8–2.0
6.3–6.6
2.0–4.0
5.1–5.7
pH = log10
1
= -log10 [H3 O+].
[H3 O+]
The hydronium ion concentration is measured in moles per liter. Vinegar has a pH of three,
distilled water a pH of 7, seawater a pH of 8.15, and household ammonia a pH of 12. The
total scale ranges from about 0.1 for normal hydrochloric acid to 14 for a normal solution
of sodium hydroxide.
Another example of the use of common logarithms is the decibel or dB (“dee bee”)
scale for measuring loudness. If I is the intensity of sound in watts per square meter, the
decibel level of the sound is
Sound level = 10 log10 sI * 10 12 d dB.
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Chapter 7: Transcendental Functions
Typical sound levels
Threshold of hearing
Rustle of leaves
Average whisper
Quiet automobile
Ordinary conversation
Pneumatic drill 10 feet
away
Threshold of pain
0 dB
10 dB
20 dB
50 dB
65 dB
90 dB
120 dB
If you ever wondered why doubling the power of your audio amplifier increases the sound
level by only a few decibels, Equation (6) provides the answer. As the following example
shows, doubling I adds only about 3 dB.
EXAMPLE 7
Sound Intensity
Doubling I in Equation (6) adds about 3 dB. Writing log for log10 (a common practice), we
have
Sound level with I doubled =
=
=
=
L
10 log s2I * 10 12 d
10 log s2 # I * 10 12 d
10 log 2 + 10 log sI * 10 12 d
original sound level + 10 log 2
original sound level + 3.
Copyright © 2005 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Eq. (6) with 2I for I
log10 2 L 0.30
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Chapter 7: Transcendental Functions
EXERCISES 7.4
Algebraic Calculations With a x and loga x
17. y = scos ud22
18. y = sln udp
Simplify the expressions in Exercises 1–4.
19. y = 7sec u ln 7
20. y = 3tan u ln 3
log5 7
1. a. 5
d. log4 16
log2 3
2. a. 2
log822
b. 9
1
f. log4 a b
4
c. plogp 7
1
f. log3 a b
9
c. log2 se sln 2dssin xd d
b. log e se x d
c. log4 s2e
e. log3 23
b. 10
log10 s1>2d
e. log121 11
log4 x
log3 x
log5 s3x 2d
4. a. 25
sin 3t
c. 1.3
b. 8
d. log11 121
3. a. 2
log1.3 75
x
sin x
d
Express the ratios in Exercises 5 and 6 as ratios of natural logarithms
and simplify.
log 2 x
log 2 x
log x a
5. a.
b.
c.
log 3 x
log 8 x
log x2 a
log 9 x
log210 x
log a b
6. a.
b.
c.
log 3 x
log22 x
log b a
Solve the equations in Exercises 7–10 for x.
21. y = 2
22. y = 5-cos 2t
23. y = log 2 5u
24. y = log 3s1 + u ln 3d
25. y = log 4 x + log 4 x 2
26. y = log 25 e x - log 5 1x
27. y = log 2 r # log 4 r
28. y = log 3 r # log 9 r
29. y = log 3 a a
x + 1
b
x - 1
31. y = u sin slog 7 ud
33. y = log 5 e x
35. y = 3log 2 t
37. y = log 2 s8t ln 2 d
ln 3
b
ln 5
a
7x
b
3x
+ 2
B
sin u cos u
b
32. y = log 7 a
e u 2u
30. y = log 5
34. y = log 2 a
x 2e 2
22x + 1
36. y = 3 log 8 slog 2 td
b
38. y = t log 3 A e ssin tdsln 3d B
Logarithmic Differentiation
7. 3log3 s7d + 2log2 s5d = 5log5 sxd
In Exercises 39–46, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.
8. 8log8 s3d - e ln 5 = x 2 - 7log7 s3xd
2
9. 3log3 sx d = 5e ln x - 3 # 10 log10 s2d
39. y = sx + 1dx
40. y = x sx + 1d
41. y = s 1td
42. y = t 2t
43. y = ssin xdx
44. y = x sin x
t
1
10. ln e + 4-2 log4 sxd = x log10 s100d
Derivatives
45. y = x
ln x
46. y = sln xdln x
In Exercises 11–38, find the derivative of y with respect to the given
independent variable.
Integration
11. y = 2x
12. y = 3-x
Evaluate the integrals in Exercises 47–56.
13. y = 52s
14. y = 2ss
15. y = x p
16. y = t 1 - e
2
d
47.
L
5x dx
48.
L
s1.3dx dx
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7.4
1
49.
0
2-u du
L0
50.
22
51.
p>2
53.
7cos t sin t dt
L0
4
55.
2x
x s1 + ln xd dx
L2
21x
dx
L1 1x
p>4
tan t
1
a b
sec2 t dt
54.
3
L0
2 ln x
2
56.
x dx
L1
4
2
x2sx d dx
L1
L-2
5-u du
52.
L
3x 23 dx
58.
L
3
59.
L0
s 12 + 1dx 12 dx
x 22 - 1 dx
x sln 2d - 1 dx
L1
Evaluate the integrals in Exercises 61–70.
4 log x
log10 x
2
dx
61.
62.
x
x dx
L1
L
4 ln 2 log x
e 2 ln 10 log x
2
10
dx
dx
63.
64.
x
x
L1
L1
2 log sx + 2d
10 log s10xd
2
10
dx
dx
65.
66.
x
x
+
2
L1>10
L0
9 2 log sx + 1d
3 2 log sx - 1d
10
2
dx
dx
67.
68.
x
+
1
x
- 1
L0
L2
69.
dx
L x log 10 x
70.
71.
L1
1>x
73.
L1
1
t dt, x 7 1
1
t dt,
x 7 0
ex
72.
a. What value of [H3 O+] minimizes the sum of the concentrations,
S = [H3 O+] + [OH -] ? (Hint: Change notation. Let
x = [H3 O+].)
b. What is the pH of a solution in which S has this minimum
value?
c. What ratio of [H3 O+] to [OH - ] minimizes S ?
T 84. Could x ln 2 possibly be the same as 2ln x for x 7 0? Graph the two
functions and explain what you see.
85. The linearization of 2x
a. Find the linearization of ƒsxd = 2x at x = 0. Then round its
coefficients to two decimal places.
T b. Graph the linearization and function together for
-3 … x … 3 and -1 … x … 1.
86. The linearization of log3 x
a. Find the linearization of ƒsxd = log 3 x at x = 3. Then round
its coefficients to two decimal places.
T b. Graph the linearization and function together in the window
0 … x … 8 and 2 … x … 4.
dx
2
L xslog 8 xd
Evaluate the integrals in Exercises 71–74.
ln x
81. In any solution, the product of the hydronium ion concentration
[H3 O+] (moles> L) and the hydroxyl ion concentration [OH -]
(moles> L) is about 10 -14 .
T 83. The equation x 2 = 2x has three solutions: x = 2, x = 4, and one
other. Estimate the third solution as accurately as you can by
graphing.
e
60.
501
82. Could log a b possibly equal 1>logb a ? Give reasons for your answer.
Evaluate the integrals in Exercises 57–60.
57.
a x and loga x
Calculations with Other Bases
1
t dt
L1
x
1
1
dt,
74.
ln a L1 t
x 7 0
T 87. Most scientific calculators have keys for log10 x and ln x. To find
logarithms to other bases, we use the Equation (5), log a x =
sln xd>sln ad .
Find the following logarithms to five decimal places.
Theory and Applications
a. log 3 8
b. log 7 0.5
75. Find the area of the region between the curve y = 2x>s1 + x 2 d
and the interval -2 … x … 2 of the x-axis.
c. log 20 17
d. log 0.5 7
e. ln x, given that log 10 x = 2.3
76. Find the area of the region between the curve y = 21 - x and the interval -1 … x … 1 of the x-axis.
77. Blood pH The pH of human blood normally falls between 7.37
and 7.44. Find the corresponding bounds for [H3 O+].
78. Brain fluid pH The cerebrospinal fluid in the brain has a hydronium ion concentration of about [H3 O+] = 4.8 * 10 -8 moles
per liter. What is the pH?
79. Audio amplifiers By what factor k do you have to multiply the
intensity of I of the sound from your audio amplifier to add 10 dB
to the sound level?
80. Audio amplifiers You multiplied the intensity of the sound of
your audio system by a factor of 10. By how many decibels did
this increase the sound level?
f. ln x, given that log 2 x = 1.4
g. ln x, given that log 2 x = -1.5
h. ln x, given that log 10 x = -0.7
88. Conversion factors
a. Show that the equation for converting base 10 logarithms to
base 2 logarithms is
log 2 x =
ln 10
log10 x .
ln 2
b. Show that the equation for converting base a logarithms to
base b logarithms is
log b x =
ln a
log a x .
ln b
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