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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