Math 3201

Chapter 7

Final Review

Multiple Choice

____

1. Predict the x-intercept of f(x) = 10 log x. A. B. C. D.

____

–1 0 1 10

2. Predict the x-intercept of f(x) = –

ln x.

A. –1 B. 0 C. 1 D. – ____

3. Predict the end behaviour of f(x) = 10 log x. A. B. C. D.

____

curve extends from quadrant I to quadrant II curve extends from quadrant I to quadrant IV curve extends from quadrant IV to quadrant I curve extends from quadrant II to quadrant I

4. Which function will have the fastest increase in the y-values? A.

y=

log x

B. y = 3 log x C. y = – log x D. y = –5 log x ____

5. Which function will have the fastest increase in the y-values? A.

y=

ln x

B. y = 9 ln x C. y= ln x D. y = 20 ln x ____

6. Which function will have the fastest decrease in the y-values? A.

y=–

log x

B. y = –2 log x C. y = –log x D. y = –5 log x ____

7. Which function will have the fastest decrease in the y-values? A.

y=–

ln x

B. y = –2 ln x C. y = –ln x D. y = –1.5 ln x ____

8. Match the following graph with its function.

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Math 3201

A.

y=–

Chapter 7

Final Review

ln x

B. y = 3 log x C. y = – (3)x D. y = 0.3(10)x ____

9. Match the following graph with its function.

A.

y=–

ln x

B. y = 3 log x C. y = – (3)x D. y = 0.3(10)x ____

10. Match the following graph with its function.

A.

y=–

ln x

B. y = 3 log x C. y = – (3)x June 2014

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Chapter 7

Final Review

D. y = 0.3(10)x ____

11. Match the following graph with its function.

A.

y=–

ln x

B. y = 3 log x C. y = – (3)x D. y = 0.3(10)x ____

12. Which exponential equation correctly represents the logarithmic equation y = log 50? A. B. C. D.

____

13. Which exponential equation correctly represents the logarithmic equation y = ln 20? A. B. C. D.

____

x = ln 7 x = ln 10 7 = ln x 10 = ln x

16. Estimate the value of y in the exponential equation 40 = 10y. A. B. C. D.

____

x = log 7 x = log 10 7 = log x 10 = log x

15. Which logarithmic equation correctly represents the exponential equation ex = 7? A. B. C. D.

____

20y = e ey = 20 y20 = e ye = 20

14. Which logarithmic equation correctly represents the exponential equation 107 = x? A. B. C. D.

____

50y = 10 10y = 50 y50 = 10 y10 = 50

0.6 1.1 1.6 2.1

17. Evaluate the logarithmic expression log16 4. A. 0 B. 0.5 C. 1 June 2014

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Math 3201

Chapter 7

Final Review

D. 2 ____

18. Evaluate the logarithmic expression log2 A. B. C. D.

____

1.3 × 10–13 mol/L 1.6 × 10–13 mol/L 1.3 × 10–12 mol/L 1.6 × 10–12 mol/L

2 40 10 000 100 000

23. What is the sound level of a noise ten times as intense as a conversation at 68 dB? Recall that sound level, β, in decibels, is defined by the equation β = 10(log I + 12) where I is the sound intensity in watts per square metre. A. B. C. D.

____

6.5 –5.2 –6.5 5.2

22. How many times greater is the intensity of 80 dB sound than the intensity of 40 dB sound? Recall that sound level, β, in decibels, is defined by the equation β = 10(log I + 12) where I is the sound intensity in watts per square metre. A. B. C. D.

____

2 3 4 5

21. Determine the concentration of hydrogen ions in bleach, with a pH of 12.8. Recall that pH, p(x), is defined by the equation p(x) = –log x where the concentration of hydrogen ions, x, in a solution is measured in moles per litre. A. B. C. D.

____

.

20. Calculate the pH of a solution with a hydrogen ion concentration of 6.5 10–6 mol/L. Recall that pH, p(x), is defined by the equation p(x) = –log x where the concentration of hydrogen ions, x, in a solution is measured in moles per litre. A. B. C. D.

____

3 0.25 –0.25 –3

19. Evaluate the logarithmic expression A. B. C. D.

____

.

680 dB 58 dB 69 dB 78 dB

24. Which expression is equivalent to

?

A. ln 8 – ln 5 B. ln 5 – ln 8 June 2014

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Math 3201

Chapter 7

Final Review

C. 8 ln 5 D. ln 0.625 ____

25. Which expression is equivalent to log 88? A. B. C. D.

____

26. Which expression is equivalent to ln 5 + ln 6? A. B. C. D.

____

3 7 1 64

32. Which value is the best estimate for x in A. B. C. D.

____

–4 4 2 0

31. Evaluate:

A. B. C. D. ____

2 log 4 2 log 8 4 log 2 8 log 2

30. Evaluate:

A. B. C. D. ____

ln 15 ln 243 ln 8 ln 125

29. Which expression is equivalent to log 64? A. B. C. D.

____

ln 39 ln 39e ln 4 ln 4e

28. Which expression is equivalent to 3 ln 5? A. B. C. D.

____

ln 11 ln 30 ln 1.2 ln 1

27. Which expression is equivalent to ln 52 – ln 13? A. B. C. D.

____

log 80 + log 8 log 22 + log 4 log 11 + log 2 log 100 – log 12

?

–0.3 0.2 0.7 1.3

33. Which value is the best estimate for x in

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Math 3201

A. B. C. D. ____

?

4.2 4.7 5.2 5.7

35. The equation of the logarithmic function that models a data set is y = 8.2 + 0.7 ln x. Determine the domain of this function. A. B. C. D.

____

Final Review

–0.05 0.05 0.95 1.05

34. Which value is the best estimate for A. B. C. D.

____

Chapter 7

{x | x ∈ R} {x | x > 0, x ∈ R} {x | x > 0.7, x ∈ R} {x | x > 8.2, x ∈ R}

36. The following data set involves logarithmic growth. Determine the missing value. x 1 5 10 20 50 100 y 0.0 0.7 1.0 1.3 1.7 A. B. C. D.

2.0 2.3 2.7 3.0

Short Answer

1. Predict whether the following logarithmic function is increasing or decreasing. f(x) = 10 log x 2. Predict whether the following logarithmic function is increasing or decreasing. f(x) = –

ln x

3. What is the location of the x-intercept for the function y = –2 log x? 4. Predict the number of y-intercepts of f(x) = 10 log x. 5. Predict the end behaviour of f(x) = 2.7 ln x. 6. Predict the end behaviour of f(x) = –

7. Predict the domain of f(x) = –

8. Predict the range of f(x) = –

log x.

ln x.

ln x.

9. Write 0.2 = ln x as an exponential equation. 10. Write x = ln 72 as an exponential equation. 11. Write 56 = 10y as a logarithmic equation. 12. Write x = 103.5 as a logarithmic equation. 13. Evaluate the logarithmic expression log7 343 without technology. June 2014

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Math 3201

Chapter 7

Final Review

14. Evaluate the logarithmic expression log2 128 without technology. 15. Evaluate the logarithmic expression log4 (–16) without technology. 16. Evaluate the logarithmic expression log5 5 + log5 25 without technology. 17. Evaluate the logarithmic expression log6 216 – log6

without technology.

18. How many times more intense is an 8.5 earthquake than an 8.0 earthquake? Round your answer to the nearest hundredth. 19. Write the following expression as the logarithm of a product.

20. Write the following expression as the logarithm of a quotient. log 56 – log 4 21. Jim has $750 in an investment that earns 4% per year, compounded annually. Determine the number of years it will take for his balance to surpass $900, to three decimal places. Use the compound interest formula:

Problem

1. In terms of hydrogen ion concentration, how much more acidic is a solution with a pH of 4.0 than a solution with a pH of 5.5? Round your answer to the nearest tenth. Show your work. Recall that pH, p(x), is defined by the equation p(x) = –log x where the concentration of hydrogen ions, x, in a solution is measured in moles per litre. 2. In terms of hydrogen ion concentration, how much more acidic is a solution with a pH of 8.4 than a solution with a pH of 11.2? Round your answer to the nearest tenth. Show your work. Recall that pH, p(x), is defined by the equation p(x) = –log x where the concentration of hydrogen ions, x, in a solution is measured in moles per litre. 3. Bleach has a pH of 12.8. Determine the pH, to the nearest tenth, of a liquid with 30 times the concentration of hydrogen ions as bleach. Show your work. Recall that pH, p(x), is defined by the equation p(x) = –log x where the concentration of hydrogen ions, x, in a solution is measured in moles per litre. 4. Milk has a pH of 6.5. Determine the pH, to the nearest tenth, of a liquid with one quarter the concentration of hydrogen ions as milk. Show your work. Recall that pH, p(x), is defined by the equation p(x) = –log x where the concentration of hydrogen ions, x, in a solution is measured in moles per litre. 5. The sound level of a radio is 52 dB. Determine the sound level, to the nearest tenth, of the radio if the intensity increases by a factor of 25. Show your work. Recall that sound level, β, in decibels, is defined by the equation β = 10(log I + 12) where I is the sound intensity in watts per square metre. 6. The sound level of a concert is 102 dB. Determine the sound level, to the nearest decibel, of the concert if the intensity increases by 0.04 W/m2. Show your work. Recall that sound level, β, in decibels, is defined by the equation β = 10(log I + 12) where I is the sound intensity in watts per square metre. June 2014

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Math 3201

Chapter 7

Final Review

7. Simplify and then evaluate each of the following logarithmic expressions to determine which expression has the greater value. Show your work. A: B: 8. Simplify and then evaluate each of the following logarithmic expressions to determine which expression has the greater value. Show your work. A:

B:

9. Simplify and then evaluate each of the following logarithmic expressions to determine which expression has the greater value. Show your work. A:

B:

10. Rado thinks he has a proof that log 25 = 1. Identify Rado’s error and finish his solution. Rado’s Proof log 25 = log 5 + log 5 = 2 log 5 = log 10 =1 11. Solve 12. Solve 13. Solve

I used the product law. I simplified the expression. I used the power law. I simplified the logarithm. , and round your answer to three decimal places. Show your work. , and round your answer to three decimal places. Show your work. , and round your answer to three decimal places. Show your work.

14. A cup of coffee cools according to the function where C(t) represents the temperature in degrees Celsius and t represents the time in minutes. The coffee will eventually reach a room temperature of 21 °C. Determine when the coffee will reach a temperature of 70 °C. Show your work. 15. $1600 is invested at 3% per year, compounded monthly. In which year after the initial investment, will the investment reach $2000? Use the compound interest formula: Show your work. 16. $2500 is invested at 2.6% per year, compounded quarterly. In which year after the initial investment, will the investment reach $3000? Use the compound interest formula: Show your work. 17. A biology laboratory starts a Petri dish with 1000 bacteria. After 30.0 hours, the population is 7250. Determine the doubling time of the bacteria algebraically using logarithms, to the nearest tenth of an hour. The doubling-time equation is where A represents the population after a period of time, A0 represents the initial population, t represents the time, and d represents the time it takes for the population to double.

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