Exponential And Logarithmic Equations Worksheet - Section 4-7 Page 2
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4-7 Exponential and Logarithmic Equations
E X A M P L E
Compound Interest
2
A certain amount of money P (principal) is invested at an annual rate r com-
pounded annually. The amount of money A in the account after t years, assum-
ing no withdrawals, is given by
n
r
n
m
1 for annual compounding.
A
P 1
P(1
r)
m
How many years to the nearest year will it take the money to double if it is
invested at 6% compounded annually?
n
To find the doubling time, we replace A in A
P(1.06)
with 2P and solve for n.
S o l u t i o n
n
2P
P(1.06)
n
Divide both sides by P.
2
1.06
n
log 2
log 1.06
Take the common or natural log of both sides.
FIGURE 2
Note how log properties are used to get n out of
n log 1.06
x
y
1.06
, y
2.
1
2
the exponent position.
4
log 2
n
log 1.06
0
20
To the nearest year.
12 years
Figure 2 confirms this result.
0
Repeat Example 2, changing the interest rate to 9% compounded annually.
M A T C H E D P R O B L E M
2
E X A M P L E
Atmospheric Pressure
3
The atmospheric pressure P, in pounds per square inch, at x miles above sea
level is given approximately by
0.21x
P
14.7e
At what height will the atmospheric pressure be half the sea-level pressure?
Compute the answer to two significant digits.
Sea-level pressure is the pressure at x
0. Thus,
S o l u t i o n
0
P
14.7e
14.7
One-half of sea-level pressure is 14.7/2
7.35. Now our problem is to find x so
0.21x
that P
7.35; that is, we solve 7.35
14.7e
for x:
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