Extra Practice – Exponentials and Logarithms
Solve the following equations (you may leave logs in your answer):
1. 3 x  6
2. 4 x  12 3. 612k  8 4. 2k 3  11 5. 4 3m1  12m2
6. 3 2m5  13 m1
7. e k 1  4
10. 10e 3 z 7  5
11. 1001.02

8. e 2 y  12
3 n

13. 2 e x  1  10

 150

14. 5 e 2 x  2  15
t
16. 100  2 10  400
17. logt  1  1
9. 2e 5a 2  8
12. 5001.05 4  200
p
 1
15.  
2
 
3
 
18. log q2  1
19. logx  2  log3x  1 20. lnp  lnp  1  ln5
22. 2 logx   log x 2  8
3 k 1
23. 1.72.13 x  24.5x
21. 10 log x 1  9
24. 3 4 log x  5
25. A colony of bacteria grows exponentially. The colony begins with 3 bacteria, but 6
hours after the beginning of the experiment, it has grown to 24 bacteria.
a) Find a formula for the number of bacteria as a function of time (in hours).
b) How long will it take the colony to reach 300 bacteria?
26. A rubber ball is dropped onto a hard surface from a height of 2 meters, and it
bounces up and down. After each bounce it rises to 90% of the height from which it fell.
a) Find a formula for h(n), the height reached after n bounces.
b) How high will the ball bounce on the 12th bounce?
c) How many bounces before the ball rises no higher than 1 cm?
27. Scientists observing owl and hawk populations collect the following data. Their
initial count for the owl population is 245 owls, and the population seems to grow by 3%
each year. They initially observe 63 hawks, and this population doubles every 10 years.
a) Find a formula for the size of the population of owls in terms of time.
b) Find a formula for the size of the population of hawks in terms of time.
c) Approximately when will the two populations be equal in number?
28. If 17% of a radioactive substance decays in 5 hours, what is its half-life?
29. Suppose a colony of bacteria is known to grow exponentially with time. At the end
of 3 hours there are 10,000 bacteria. At the end of 5 hours there are 40,000. How many
bacteria were present initially?
30. The total number of hamburgers sold by a national fast-food chain is growing
exponentially. If 4 billion had been sold by 2001 and 6 billion had been sold by 2006,
how many will have been sold by 2011?
Extra Practice – Exponentials and Logarithms – Answers
1. x 
5. m 
9. a 
log 6
log 3
2. x 
log576
 16 
log 
 3 
 243 
log

13 

6. m 
 9 
log 
 13 
ln 4  2
5
log 12
log 4
10. z 
2
4 log 
 5   75
12. p 
log1.05 
3. k 
1  log 8 
log11
1 
 4. k  3 
2  log 6 
log 2
7. k  1  ln 4
 1.5
log
3

1.02

11. n 
log1.02
7  ln 2
3
14. x 
13. x  ln 4
ln 5
2
17. t  11
21. x  8
 2 
log

1 .7 

 0.225
22. x  100 23. x 
 2.13 

log

4
.
5


25a. f t   3  2
18. q   10
25b. t 
26a. hn  20.9 
n
12
27a. Owls t   2451.03
 1
5 log 
 2   1.96hrs.
28. h 
log0.17 
29. 1250 bacteria
30. 9 billion hamburgers
log 6
log 8
5
20. p  
4
2
29
1
3 4
 log

log 5 

24. x  10




4
log 2
26b. h12  20.9   0.56m
t



  17.5
15. k  
19. x 
16. t  20
t
2
8. y  2  ln 12
27b. Hawks t   63  2
t
10
log( 0.005)
 50
log0.9 
 245 
log

63 

27c. t 
 34 yrs.
 2 0 .1 

log
 1.03 
26c. n 