Section 5.5 – Exponential and Logarithmic Models
1. A savings account is opened with $20,000 earning 10.5%
compounded continuously.
a. How many years does it take for the money to double?
A  Pe
0.105 t
40000  20000e
rt
2e
ln 2  0 .1 0 5 t ln e
0.105 t
ln 2  0 .1 0 5 t
ln 2
 t
0 .1 0 5
b. How much is in the account after ten years?
A  Pe
rt
A  20000e
0.105 (10 )
A  $ 5 7 1 5 3 .0 2
t  6 .6 0 1
2. A savings account is opened with $600. At the end of ten
years, the account will have $19205. If the account earns
interest compounded continuously:
a. What is the annual percentage rate of interest?
A  Pe
rt
ln
19205  600e
19205
 e
ln
600
 1 0r ln e
600
10 r
10r
19205
r  0 .3 4 6 6 0
3 4 .6 6 0 %
19205
600
10
r
b. How many years did it take for the account to double in
value?
A  Pe
1200  600e
2e
ln 2  0 .3 4 6 6 0 t ln e
rt
0.34660 t
0.34660 t
ln 2
 t
0 .3 4 6 6 0
t  2 .0 0 0

3. Given A  P  1 

r 

n
nt
determine the principal P that must be
invested that must be invested at 12% so that $500,000 will
be available in 40 years.
500000  P 1  0.12 
3 1 0 2 7 6 .9 4
40
4. The population P of a city is P  2 4 0 3 6 0 e 0 .0 1 2 t where t = 0
represents the year 2000. According to this model, when
will the population reach 275000?
275000  240360e
275000
ln
240360
275000
e
0.012 t
0 .0 1 2 t
 0 .0 1 2 t ln e
240360
ln
275000
240360  t
0 .0 1 2
2012
5. The population P of a city is P  1 4 0 5 0 0 e k t where t = 0
represents the year 2000. In 1960, the population was
100,250. Find the value of k, and use this result predict the
population in the year 2020.
100250  140500e
100250
ln
ln
140500
100250
e
k  40 
  4 0k ln e
140500
100250
140500  k
40
k  0 .0 0 8 4 3 8 5 1 0 6
k   40 
P  140500e
P  140500e
0.008435106 t
0.008435106  20 
P  1 6 6 3 3 0 .6 5
6. The number of bacteria N in a culture is modeled by
kt
N  1 0 0 e , where t is the time in hours. If N = 300 when t = 5,
estimate the time required for the population to double in size.
N  100e
300  100e
kt
N  100e
k 5 
2e
ln 3  5k ln e
0 .2 1 9 7 2 2 4 5 7 7  k
0.21972245 77 t
0.2197224 5 7 7 t
ln 2  0 .2 1 9 7 2 2 4 5 7 7 t ln e
t  3 .1 5 5
7. A car that cost $22,000 new has a book value of $13,000 after
2 years.
a. Find the straight-line model V = mt + b
kt
b. Find the exponential model V  a e
V  ae
V  mt  b
22000  m ( 0 )  b
22000  a e
22000  b
22000  a
13000  m 2   22000
V  4500 t  22000
k(0 )
V  22000 e
V  m t  22000
m  4500
V  22000 e
kt
13 0 00  2200 0 e
ln
ln
13000
kt
k2
 2 k ln e
22000
13000
22000  k
2
k   0 .2 6 3
 0 .263 t
7. A car that cost $22,000 new has a book value of $13,000 after
2 years.
c. Use a graphing utility to graph the two models in the same
viewing window. Which model depreciates faster in the first two
years?
d. Find the book values of the car after 1 year and after 3 years
using each model.
V  4500 t  22000
V  22000 e
 0 .263 t
8. The sales S (in thousands of units) of a new product after it has
kt
been on the market t years are modeled by
S  t   1 0 0 1  .e
15000 units of the new product were sold the first year.
a. Complete the model by solving for k.
b. Use the model to estimate the number of units sold after five
years.


S  t   100 1  e

15  100 1  e
0 .1 5  1  e
0 .8 5  e
kt

k  1

S  t   100 1  e


S 5   100 1  e
k  1
k
ln 0 .8 5  k ln e
k   0 .1 6 2 5 1 8 9 2 9 5
55
 0 .1 6 2 5 t

 0 .1 6 2 5  5 


9. The sales S (in thousands of units) of a product after x hundred
dollars is spend on advertising are modeled by S  1 0 1  e k x .
When $500 is spend on advertising, 2500 units are sold.
a. Complete the model by solving for k.
b. Estimate the number of units that will be sold if advertising
expenditures are raised to $700.



5k
2 .5  1 0  1  e 
S  10 1  e
0 .2 5  1  e
0 .7 5  e
5k
5k

 0 .0 5 7 5 x

 0 .0 5 7 5  7 
S  10 1  e
S  10 1  e
5k
ln 0 .7 5  5k ln e
ln 0 .7 5
k
5
k   0 .0 5 7 5 3 6 4 1 4 5

3315


10. The management at a factory has found that the maximum
number of units a worker can produce in a day is 30. The
learning curve for the number of units N produced per day after a
new employee has worked t days is N  3 0 1  e k t . After 20 days
on the job, a new employee produces 19 units.
a. Find the learning curve for this employee (first, find the value
of k).
b. How many days should pass before this employee is
producing 25 units per day?
c. Is the employee’s production increasing at a linear rate?
Explain your reasoning.


19  30 1  e
19
 1 e
30
11
30
e
2 0k
2 0k
20k

ln
11

 2 0k ln e
30
ln
11
30  k
20

N  30 1  e
 0 .0 5 0 2 t

10. The management at a factory has found that the maximum
number of units a worker can produce in a day is 30. The
learning curve for the number of units N produced per day after a
new employee has worked t days is N  3 0 1  e k t . After 20 days
on the job, a new employee produces 19 units.
a. Find the learning curve for this employee (first, find the value
of k).
b. How many days should pass before this employee is
producing 25 units per day?
c. Is the employee’s production increasing at a linear rate?
Explain your reasoning.
5


 0 .0 5 0 2 t

 0 .0 5 0 2 t
N  30 1  e
25  30 1  e
25
30
 1 e
 0 .0 5 0 2 t


5
 e
 0 .0 5 0 2 t
30
ln
5
30
  0 .0 5 0 2 t ln e

ln
30
 t
 0 .0 5 0 2
t  3 5 .7 1 7
11. On the Richter scale, the magnitude R of an earthquake of
intensity I is R  lo g1 0
I
I0
where I0  1 is the minimum
intensity used for comparison. Find the magnitude R of an
earthquake of intensity I if:
R  log10 80500000  7.906
a. I = 80500000
R  log10 48275000  7.684
b. I = 48275000
R  log10 251200  5.400
c. 251,200
R  lo g1 0
I
1
R  log10 I