Exponents and Logarithms Exam Questions · x log y 3 1 2log as a single logarithm is: a. 3 2 log y...

Exponents and Logarithms

Exam Questions

Name: ANSWERS

2 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. 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


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


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


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


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)


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


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


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)


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)


29. a. Sketch the graph of xy ln . (2 marks)

b. Sketch the graph of 2ln xy . (2 marks)


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)


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)


34. a. Sketch the graph of 13 xxf . (2 marks)

b. Sketch the graph of xf 1 . (1 mark)


35. Given 129.19log a and 712.04log a , find the value of 12log a . (3 marks)

(calculator)


1log5log1log2 222 xxx


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)


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)


39. Sketch the graphs of:

a.

x

y

4

1 (2 marks)

b.

x

y

4

12 (1 mark)


40. Evaluate: (3 marks)

3log24log144log2

1333


xx 444 log3log2log


42. Identify which of these values is greater. Justify your answer. (1 mark)

80log5 30log3

43. Solve: 35 532

xx (calculator) (4 marks)


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)


46. a. Sketch the graph of 1log5 xxf . (2 marks)

b. Sketch the graph of xf 1 . (1 mark)


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


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)


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


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


decimal) a (as rateinterest annuali

n = the number of equal periodic payments. Express your answer as a whole number. (3 marks)


56. Using the laws of logarithms, fully expand the expression: (3 marks)

1log

3

2y

xw


25log3log 33 xx


58. Solve: xx 279 12 (3 marks)

59. Expand using the laws of logarithms. (2 marks)

4

logb

a


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


periodeach amount annual thei

n = the number of investments


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)


63. Sketch the graph of 22 xy . (3 marks)

64. Explain why the domain of 1log 2 xy is 1x . (1 mark)


65. If p6log , r5log and q2log , express 60log in terms of p , q , and r .

(2 marks)

Exponents and Logarithms Exam Questions · x log y 3 1 2log as a single logarithm is: a. 3 2 log y...

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