X - UC Santa Barbara
Extra Practice ? Exponentials and Logarithms
Solve the following equations (you may leave logs in your answer): 1. 3 x 6 2. 4x 12 3. 612k 8 4. 2k3 11 5. 43m1 12m2
6. 32m5 13m1
7. ek1 4 8. e2y 12
9. 2e5a2 8
10. 10e3z7 5
13. 2 ex 1 10
11. 1001.023n 150
14. 5 e2x 2 15
t
16. 100 210 400
17. logt 1 1
12.
5001.05
p 4
200
15. 1 3k1 3 2
18. log q2 1
19. logx 2 log3x 1 20. lnp lnp 1 ln5 21. 10logx1 9
22. 2logx log x2 8
23. 1.72.13x 24.5x
24. 34 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 log6 log3
2. x log12 log 4
3.
k
1 2
1
log8 log 6
4. k 3 log11 log 2
5. m log576
log
16 3
6.
m
log
243 13
log
9 13
7. k 1 ln 4
8. y 2 ln12
9. a ln 4 2 5
10. z 7 ln2 3
11.
n
log
1.5
1.023
log1.02
17.5
12.
p
4
log
2 5
log1.05
75
13. x ln 4
14. x ln5 2
15. k log6 log8
16. t 20 17. t 11 18. q 10
19. x 2 20. p 5
29
4
21. x 8
22. x 100
log 2
23. x 1.7 0.225
log
2.13
4.5
1
24.
x
10
log 3 log 5
4
t
25a. ft 3 2 2
25b. t 4 log 2
26a. hn 20.9n 26b. h12 20.912 0.56m
26c.
n
log(0.005)
log0.9
50
t
27a. Owlst 2451.03t 27b. Hawkst 63 210
log 245
27c. t 63 34yrs.
log
20.1 1.03
28.
h
5
log
1 2
log0.83
18.6hrs.
29. 1250 bacteria 30. 9 billion hamburgers
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