Exponents and Logarithms Exam Questions Name: ANSWERS
3 January 2013 – January 2017
Multiple Choice
1. If 4log 2 x , then x2log 2 is equal to:
a. 5 b. 8 c. 16 d. 32
2. Identify the value of the x -intercept of the function 2ln xy .
a. -1 b. 0 c. 2 d. 3
3. Which equation is represented by the graph sketched below.
a.
x
y
2
1
b.
x
y
2
1
c. xy 2
d. xy 2
4 January 2013 – January 2017
4. The graph of 62log 2 xy intersects the graph of 4y at:
a. 1x b. 1x c. 5x d. 14x
5. The graph of
x
y
2
1compared to the graph of
y
x
2
1 is a:
a. reflection in the x -axis b. reflection in the y -axis
c. reflection in the line xy d. reciprocal function
6. The graph of the function xf shown below is best described by the equation:
a. 32 xxf b. 32 xxf c. 32 xxf d. 32 xxf
5 January 2013 – January 2017
7. Which of the following is a reasonable estimate for the value of 350log ?
a. 2 b. 2.5 c. 2.8 d. 3
8. Solve: 75ln xe
a. -2 b. –ln2 c. ln7 – ln5 d. 5
7
9. Simplify the following expression: 2log36log2
1aa
a. 3log a b. 4log a c. 9log a d. 12log a
10. Which of the following is closest to the value of 125log40log 52 ?
a. 3 b. 8 c. 10 d. 45
6 January 2013 – January 2017
11. The -x intercept of the graph of 13 xy is:
a. -1 b. 0 c. 1 d. 2
12. The expression yx log3
1log2 as a single logarithm is:
a. 3
2
logy
x b.
y
x
3
2log c. 32log yx d. 32log yx
13. Determine the value of 27loglog 39 .
a. 3
1 b.
2
1 c. 2 d. 3
14. Identify an equivalent expression for 5log1 2 .
a. 5log 2 b. 7log 2 c. 10log 2 d. 11log 2
7 January 2013 – January 2017
15. Solve: x2log77
a. 1x b. 2x c. 7x d. 49x
16. Identify the logarithmic form of 65 x .
a. 6log5 x b. x6log5 c. 5log6 x d. x5log6
8 January 2013 – January 2017
Written Response
17. Given 3log ab , give one example of possible values of a and b that make this equation true.
(1 mark)
18. Frank tried to expand a logarithmic expression using the laws of logarithms. He made one error.
Frank’s solution:
wzxzw
xaaaaa loglog2loglog
2log
Write the correct solution. (1 mark)
19. Estimate the value of 35log5 .
Justify your answer. (1 mark)
9 January 2013 – January 2017
20. Claire correctly solves the following equation:
23log6log 22 xx .
She finds two possible values of x : 2x and 7x . Identify which one of these values is unacceptable and explain why. (1 mark)
21. Which expression has a larger value?
80logor 36log 32
Justify your answer. (1 mark)
22. Which of the following equations could be solved without the use of logarithms? Without actually solving the problem, explain your choice. (1 mark)
13104 xx or 14
12
273
1
x
x
10 January 2013 – January 2017
23. Using the laws of logarithms, expand: (2 marks)
z
yxalog
24. Determine the interceptx and intercepty of 14logy 2 x . (2 marks)
25. Solve the following equations algebraically: (3 marks)
12log4log 33 xx
11 January 2013 – January 2017
26. a. Sketch the graph of xy 3 . (2 marks)
b. Explain how the graph of xy 3 can be used to sketch the graph of xy 3log .
(1 mark)
12 January 2013 – January 2017
27. Determine the value of y in the following equation: (3 marks)
yxxx log23log27log
28. The number of times a website is visited can be modeled by the function:
rteA 800
where A = the total number of visitors at time t
t = the time in days 0t
r = the rate of growth
After 5 days, 40 000 people have visited the site. Determine the number of visitors expected after 9 days. Express your answer as a whole number. (calculator) (3 marks)
13 January 2013 – January 2017
29. a. Sketch the graph of xy ln . (2 marks)
b. Sketch the graph of 2ln xy . (2 marks)
14 January 2013 – January 2017
30. Solve algebraically: (calculator) (3 marks)
53 710 xx
31. Jess invests $12 000 at a rate of 4.75% compounded monthly. How long will it take for Jess to triple her investment? Express your answer in years, correct to 3 decimal places. (3 marks) (calculator)
15 January 2013 – January 2017
32. Solve the following equation: (4 marks)
13loglog2 44 xx
33. An earthquake in Vancouver had a magnitude of 6.3 on the Richter scale. An earthquake in Japan had a magnitude of 8.9 on the Richter scale. How many times more intense was the Japan earthquake than the Vancouver earthquake? You may use the formula below:
0
logA
AM
where M is the magnitude of the earthquake on the Richter scale A is the intensity of the earthquake
0A is the intensity of a standard earthquake
Express your answer as a whole number. (calculator) (2 marks)
16 January 2013 – January 2017
34. a. Sketch the graph of 13 xxf . (2 marks)
b. Sketch the graph of xf 1 . (1 mark)
17 January 2013 – January 2017
35. Given 129.19log a and 712.04log a , find the value of 12log a . (3 marks)
(calculator)
36. Solve the following equation: (4 marks)
1log5log1log2 222 xxx
18 January 2013 – January 2017
37. Determine how many monthly investments of $50 would have to be deposited into a savings account that pays 3% annual interest, compounded monthly, for the account’s future value to be $50,000.
Use the formula: i
iRFV
n11
where: FV = the future value
R = the investment amount
yearper periods gcompoundin ofnumber the
rateinterest annual thei
n = the number of investments Express your answer as a whole number. (calculator) (3 marks)
19 January 2013 – January 2017
38. A population of 500 bacteria will triple in 20 hours. (calculator) Using the formula given below,
rtPeA
A = population after t hours
P = initial population r = rate of growth t = time in hours a. Determine the rate of growth r . (2 marks) b. Determine how many hours it will take for the initial population to double with the same rate of growth. (2 marks)
21 January 2013 – January 2017
40. Evaluate: (3 marks)
3log24log144log2
1333
41. Solve the following equation: (3 marks)
xx 444 log3log2log
22 January 2013 – January 2017
42. Identify which of these values is greater. Justify your answer. (1 mark)
80log5 30log3
43. Solve: 35 532
xx (calculator) (4 marks)
23 January 2013 – January 2017
44. A lake affected by acid rain has a pH of 4.4. A person suffering from heartburn has a stomach acid of pH of 1.2.
The pH of a solution is defined as H-logpH where H is the hydrogen ion
concentration. How many times greater is the hydrogen ion concentration of the stomach than that of the lake? Express your answer as a whole number. (calculator) (2 marks)
45. Solve: 13loglog2 44 xx (4 marks)
24 January 2013 – January 2017
46. a. Sketch the graph of 1log5 xxf . (2 marks)
b. Sketch the graph of xf 1 . (1 mark)
25 January 2013 – January 2017
47. Kim solved the following logarithmic equation:
Explain why 3x is an extraneous root. (1 mark)
48. Solve: xx 223
956 (calculator) (4 marks)
Solution:
088.0
9log5log3
6log5log29log2
6log5log29log29log5log3
6log5log29log29log5log3
9log9log25log25log36log
9log25log236log
9log5log6log
9log56log
223
223
x
x
x
xx
xx
xx
xx
xx
26 January 2013 – January 2017
49. Evaluate: 2log 4 (1 mark)
Solution:
2
1
50. Estimate the value of 5log 2 . Justify your answer. (1 mark)
Solution:
38log
24log
2
2
therefore 3.25log 2
51. Sketch the graph of 1log3 2 xxf . (3 marks)
27 January 2013 – January 2017
52. Solve: a333 log8log3
12log4 (3 marks)
Solution:
8
log8log
log2
16log
log2log16log
log8log2log
33
33
323
33
1
3
4
3
a
a
a
a
a
53. Use the law of logarithms, fully expand the expression: (3 marks)
zy
xa
3
log
Solution:
zyx
zyxzy
x
aaa
aaaa
log2
1loglog3
loglogloglog 2
1
33
54. Given 12 xxf , state the equation of the horizontal asymptote. (1 mark)
Solution:
1y
28 January 2013 – January 2017
55. Sheeva’s bank is lending her $50 000 at an annual interest rate of 6%, compounded monthly, to purchase a car. Given that the last payment will be a partial payment, determine how many full monthly payments of $800 Sheeva will have to make. The formula below may be used.
i
iRPV
n
11
where PV = the present value of the amount borrowed R = the amount of each periodic payment
yearper periods gcompoundin ofnumber the
decimal) a (as rateinterest annuali
n = the number of equal periodic payments. Express your answer as a whole number. (3 marks)
29 January 2013 – January 2017
56. Using the laws of logarithms, fully expand the expression: (3 marks)
1log
3
2y
xw
57. Solve the following equation: (3 marks)
25log3log 33 xx
30 January 2013 – January 2017
58. Solve: xx 279 12 (3 marks)
59. Expand using the laws of logarithms. (2 marks)
4
logb
a
31 January 2013 – January 2017
60. Peter invests $560 per month at an annual interest rate of 4.2%, compounded monthly. Determine how many monthly investments he will need to make to obtain at least $500 000. Express your answer as a whole number. (calculator) (3 marks) Use the formula:
i
iRFV
n11
where FV = the future value
R = the investment amount each period
yearper periods gcompoundin ofnumber the
periodeach amount annual thei
n = the number of investments
32 January 2013 – January 2017
61. Solve the following equation algebraically: (2 marks)
3log1log5log 22 xx
62. Justify why 4.7 is a better estimate than 4.3 for the value of 26log 2 . (1 mark)
33 January 2013 – January 2017
63. Sketch the graph of 22 xy . (3 marks)
64. Explain why the domain of 1log 2 xy is 1x . (1 mark)