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CHAPTER 5
Logarithmic, Exponential,
and Other Transcendental Functions
Section 5.1

The Natural Logarithmic Function: Differentiation . . . . 493

Section 5.2

The Natural Logarithmic Function: Integration . . . . . . 498

Section 5.3

Inverse Functions . . . . . . . . . . . . . . . . . . . . . . 503

Section 5.4

Exponential Functions: Differentiation and Integration . . 509Section 5.5

Bases Other than e and Applications . . . . . . . . . . . . 516

Section 5.6

Differential Equations: Growth and Decay . . . . . . . . . 522

Section 5.7

Differential Equations: Separation of Variables

Section 5.8

Inverse Trigonometric Functions: Differentiation . . . . . 535

Section 5.9

Inverse Trigonometric Functions: Integration

. . . . . . 527

. . . .. . . 539

Section 5.10 Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . 543
Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548
Problem Solving

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554

CHAPTER 5
Logarithmic, Exponential, and Other Transcendental Functions
Section 5.1

The Natural Logarithmic Function: DifferentiationSolutions to Even-Numbered Exercises
2. (a)

(b)

y

3

3
0.2

4

2
1
−4
x
1

2

3

4

The graphs are identical.

−1

4. (a) ln 8.3
8.3

(b)
1

2.1163

0.6

1
dt
t

8. f x

6. (a) ln 0.6

2.1163

(b)
1

ln x

10. f x

0.5108

1
dt
t

0.5108

ln

x

Reflection in the x-axis

Reflection in the y-axis and the x-axis

Matches(d)

Matches (c)

12. f x

14. f x

2 ln x

ln x

16. g x

Domain: x

Domain: x > 0
y

0

ln x

Domain: x > 0
y

y
3

2

2

3

2
1
1

2

x
1

2

3

−3

4

−2

x

−1

1

2

3

1

−1
−2
−2

x

−3

ln 1
4

ln 1

2 ln 2

3 ln 2

ln 3

(c) ln

3

(d) ln

1
72

1
3

12

2 ln 2

ln 1

ln 3

3 ln 222. ln xyz

ln x

ln y

26. ln 3e2

ln 3

2 ln e

1
2

ln a

3

20. ln 23

ln 23

2

ln 2

0.8283

2 ln 3

4.2765
24. ln a

ln z

2

1.3862

3
2

2

3.1779

18. (a) ln 0.25
(b) ln 24

1

ln 3

28. ln

1
e

1

ln a

ln 1

ln e

1

12

1

1

493

494

Chapter 5

30. 3 ln x

Logarithmic, Exponential, and OtherTranscendental Functions

2 ln y

ln x3

4 ln z

ln

32. 2 ln x

34.

ln x

3
ln x2
2

36.

1

1

ln x

ln x

ln z4

x 3y 2
z4

1

1

ln y 2

2 ln

ln x

1

x
1x

x

1

3
x2 1
ln
2
x 1x 1

38. lim ln 6

3

ln

x →6

x

2

x2

1
x2
x2

ln

1
1

3

x

40. lim ln

x

x →5

x

4

f=g
−1

5
−1

42. y

ln x33
ln x
2

2

3
2x

y

3
2.

y
dy
dx

2x2

4x
2x2 1
1
ln x2
2

4

1 2x
2 x2 4

ln t
t2

ln

3

x
x

1
1
3x 1

1
x

fx

y
dy
dx

ln t

ln x

1

2x

1
x

4

1

x

ln

56.

t1 t

1
2.

x ln x

52. f x

4

x
x2

y
dy
dx

ln t
t

ht

y

48.

4x

1

ln x2

54. h t

58. y

1

1

hx

1ln x
2

2

At 1, 0 , y

ln 2x2

46. h x

ln x1
1
2x

y

At 1, 0 , y

50.

44. y

x

ln x

ln 2x

3
1
x

ln x

3

3
3

xx

3

ln ln x
1x
ln x

1
x ln x

t2
1
ln x
3

1
1
1
x

1

1

ln x

12
3 x2 1

1

60. f x
fx

2
2

3x

1

ln x
1
4

x
1
4

x2

4

x2

x2

1

x
4

x2

ln 5

1.6094

Section5.1
x2
2x2

y

62.

4

2x2 x

dy
dx

x2
x

4

4x x2

4

1
2
ln
4

x2

4
4x 4

x2
2x2
1
42

The Natural Logarithmic Function: Differentiation

4

1
x2

1
ln 2
4

x2

x
x2

4

1
ln x
4

4

4

1
4x

x
x2

4

Note that:
1
x2

2

Hence,

4

1
x2

2

4

2

2x x

4

1

12 2
2x x2
x2

4x x

64. y...