Exponential And Logarithmic Equations Worksheet - Section 4-7 Page 9
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4 INVERSE FUNCTIONS; EXPONENTIAL AND LOGARITHMIC FUNCTIONS
x
2x
Evaluate Problems 43–48 to four decimal places.
69. e
x
0
70. xe
1
0
x
x
71. xe
2
0
72. e
2x
0
43. log
372
44. log
23
45. log
0.0352
5
4
8
2
73. ln x
2x
0
74. ln x
x
0
46. log
0.005 439 47. log
0.1483
48. log
435.62
2
3
12
x
75. ln x
e
0
76. ln x
x
0
C
APPLICATIONS
Solve Problems 49–56 for the indicated variable in terms of
the remaining symbols. Use the natural log for solving
Solve Problems 77–90 algebraically or graphically, whichever
exponential equations.
seems more appropriate.
rt
49. A
Pe
for r (finance)
77. Compound Interest. How many years, to the nearest
year, will it take a sum of money to double if it is invested
nt
r
50.
A
P 1
for t (finance)
at 15% compounded annually?
n
78. Compound Interest. How many years, to the nearest
I
51.
D
10 log
for I (sound)
year, will it take money to quadruple if it is invested at
I
0
20% compounded annually?
1
79. Compound Interest. At what annual rate compounded
52.
t
(ln A
ln A
) for A (decay)
0
k
continuously will $1,000 have to be invested to amount to
$2,500 in 10 years? Compute the answer to three signifi-
I
53.
M
6
2.5 log
for I (astronomy)
cant digits.
I
0
80. Compound Interest. How many years will it take $5,000
54. L
8.8
5.1 log D for D (astronomy)
to amount to $8,000 if it is invested at an annual rate of
E
9% compounded continuously? Compute the answer to
Rt/L
55.
I
(1
e
) for t (circuitry)
three significant digits.
R
81. Astronomy. The brightness of stars is expressed in terms
n
(1
i)
1
56.
S
R
for n (annuity)
of magnitudes on a numerical scale that increases as the
i
brightness decreases. The magnitude m is given by the
formula
The following combinations of exponential functions define
L
four of six hyperbolic functions, an important class of
m
6
2.5 log
L
functions in calculus and higher mathematics. Solve Problems
0
57–60 for x in terms of y. The results are used to define inverse
where L is the light flux of the star and L
is the light flux
0
hyperbolic functions, another important class of functions in
of the dimmest stars visible to the naked eye.
calculus and higher mathematics.
(A) What is the magnitude of the dimmest stars visible to
the naked eye?
x
x
x
x
e
e
e
e
57.
y
58.
y
2
2
(B) How many times brighter is a star of magnitude 1
than a star of magnitude 6?
x
x
x
x
e
e
e
e
59.
y
60.
y
82. Astronomy. An optical instrument is required to observe
x
x
x
x
e
e
e
e
stars beyond the sixth magnitude, the limit of ordinary vi-
sion. However, even optical instruments have their limita-
In Problems 61–64, use a graphing utility to graph each
tions. The limiting magnitude L of any optical telescope
function. [Hint: Use the change-of-base formula first.]
with lens diameter D, in inches, is given by
61. y
3
log
(2
x)
62. y
log
(4
x)
5
L
8.8
5.1 log D
2
3
63. y
log
x
log
x
64. y
log
x
log
x
(A) Find the limiting magnitude for a homemade 6-inch
3
2
3
2
reflecting telescope.
In Problems 65–76, use a graphing utility to approximate to
(B) Find the diameter of a lens that would have a limiting
two decimal places any solutions of the equation in the interval
magnitude of 20.6.
0
x
1. None of these equations can be solved exactly
Compute answers to three significant digits.
using any step-by-step algebraic process.
83. World Population. A mathematical model for world pop-
x
x
65. 2
2x
0
66. 3
3x
0
ulation growth over short periods of time is given by
x
x
rt
67. x3
1
0
68. x2
1
0
P
P
e
0
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