Mrs. Revelle



Review Problems Unit 8 Test

Multiple Choice

Identify the choice that best completes the statement or answers the question.

____ 1. What are the parent function points for [pic]

|a. |( 1 , 0 ) and ( 1 , 1 ) |c. |( 1 , 0 ) and ( 7 , 1 ) |

|b. |( 0 , 1 ) and ( 1 , 10 ) |d. |( 1 , 0 ) and ( 10 , 1 ) |

Write the equation in logarithmic form.

____ 2. [pic]

|a. |[pic] |c. |[pic] |

|b. |[pic] |d. |[pic] |

____ 3. [pic]

|a. |[pic] |c. |[pic] |

|b. |[pic] |d. |[pic] |

Evaluate the logarithm.

____ 4. [pic]

|a. |–3 |b. |5 |c. |–4 |d. |4 |

____ 5. [pic]

|a. |5 |b. |–5 |c. |4 |d. |3 |

____ 6. log 0.01

|a. |–10 |b. |–2 |c. |2 |d. |10 |

____ 7. Write the equation [pic] in exponential form.

|a. |[pic] |b. |[pic] |c. |[pic] |d. |[pic] |

Graph the logarithmic equation.

____ 8. [pic]

|a. |[pic] |c. |[pic] |

|b. |[pic] |d. |[pic] |

____ 9. [pic]

|a. |[pic] |c. |[pic] |

|b. |[pic] |d. |[pic] |

____ 10. [pic]

|a. |[pic] |c. |[pic] |

|b. |[pic] |d. |[pic] |

____ 11. [pic]

|a. |[pic] |c. |[pic] |

|b. |[pic] |d. |[pic] |

Write the expression as a single logarithm.

____ 12. [pic]

|a. |[pic] |c. |[pic] |

|b. |[pic] |d. |[pic] |

____ 13. [pic]

|a. |[pic] |b. |[pic] |c. |[pic] |d. |[pic] |

____ 14. [pic]

|a. |[pic] |c. |[pic] |

|b. |[pic] |d. |none of these |

Expand the logarithmic expression.

____ 15. [pic]

|a. |[pic] |c. |[pic] |

|b. |[pic] |d. |[pic] |

____ 16. [pic]

|a. |[pic] |c. |[pic] |

|b. |[pic] |d. |[pic] |

____ 17. [pic]

|a. |[pic] |c. |[pic] |

|b. |[pic] |d. |[pic] |

____ 18. Use the properties of logarithms to evaluate [pic].

|a. |2 |b. |4 |c. |8 |d. |41 |

____ 19. Solve [pic]. Round to the nearest ten-thousandth.

|a. |0.6616 |b. |2.6466 |c. |1.7509 |d. |1.9091 |

____ 20. Solve [pic].

|a. |[pic] |b. |[pic] |c. |[pic] |d. |[pic] |

____ 21. Solve [pic].

|a. |–1.8847 |b. |–0.1069 |c. |0.3375 |d. |1.0378 |

____ 22. Use a graphing calculator. Solve [pic] by graphing. Round to the nearest hundredth.

|a. |1.19 |b. |0.83 |c. |4.76 |d. |3.33 |

____ 23. Solve [pic].

|a. |[pic] |b. |[pic] |c. |[pic] |d. |[pic] |

____ 24. Solve [pic]. Round to the nearest ten-thousandth.

|a. |10.7722 |b. |5 |c. |2.7826 |d. |0.6309 |

____ 25. Solve [pic]. Round to the nearest hundredth if necessary.

|a. |0.33 |b. |0.04 |c. |3 |d. |27 |

____ 26. Solve [pic].

|a. |0.0090 |b. |0.3103 |c. |3.2222 |d. |111 |

____ 27. Solve [pic].

|a. |12.3308 |b. |43.3013 |c. |86.6025 |d. |1875 |

Write the expression as a single natural logarithm.

____ 28. [pic]

|a. |[pic] |b. |[pic] |c. |[pic] |d. |[pic] |

____ 29. [pic]

|a. |[pic] |b. |[pic] |c. |[pic] |d. |[pic] |

____ 30. [pic]

|a. |[pic] |b. |[pic] |c. |[pic] |d. |[pic] |

____ 31. Simplify [pic].

|a. |3 |b. |[pic] |c. |3e |d. |[pic] |

____ 32. Solve [pic]. Round to the nearest thousandth.

|a. |1,489.979 |b. |2,979.958 |c. |2,981.458 |d. |1,490.979 |

____ 33. Solve [pic].

|a. |50,000 |b. |74.2 |c. |10 |d. |3 |

____ 34. Solve [pic].

|a. |6 |b. |6e |c. |[pic] |d. |ln 6 |

Use natural logarithms to solve the equation. Round to the nearest thousandth.

____ 35. [pic]

|a. |–0.448 |b. |0.327 |c. |0.067 |d. |–0.046 |

____ 36. [pic]

|a. |–0.288 |b. |–0.275 |c. |0.275 |d. |0.288 |

____ 37. [pic]

|a. |–1.664 |b. |0.073 |c. |0.168 |d. |0.190 |

____ 38. Graph the relation and its inverse. Use open circles to graph the points of the inverse.

|x |0 |4 |9 |10 |

|y |3 |2 |7 |–1 |

|a. |[pic] |c. |[pic] |

|b. |[pic] |d. |[pic] |

____ 39. Find the inverse of [pic].

|a. |[pic] |c. |[pic] |

|b. |[pic] |d. |[pic] |

____ 40. Graph [pic] and its inverse.

|a. |[pic] |c. |[pic] |

|b. |[pic] |d. |[pic] |

____ 41. For the function [pic], find [pic]. Determine whether [pic] is a function.

|a. |[pic]; [pic] is not a function. |

|b. |[pic]; [pic] is not a function. |

|c. |[pic]; [pic] is a function. |

|d. |[pic]; [pic] is a function. |

Essay

42. Suppose you invest $580 at 10% compounded continuously.

a. Write an exponential function to model the amount in your investment account.

b. Explain what each value in the function model represents.

c. In how many years will the total reach $3600? Show your work.

43. The function [pic] models the kindergarten population y of a certain elementary school x years after the year 2000. Graph the function on your graphing calculator. Explain how to use the calculator to estimate when the kindergarten population will reach 142 and state the year that you estimate.

44. The formula [pic] gives the average atmospheric pressure P in pounds per square inch, at an altitude x in miles above sea level.

a. Find the elevation at which the average atmospheric pressure is 8.4 lb/in.2. Show the steps you used to solve this problem.

b. What is the average atmospheric pressure at sea level? Explain.

45. Consider the relation s given by the values in the table.

|x |–5 |–3 |–1 |1 |

|y |–6 |–2 |–2 |–6 |

a. Find the inverse of relation s

b. Graph s and its inverse.

c. Describe the relationship between the line y = x and the graphs of s and its inverse.

d. Is the relation s a function? How do you know?

e. Is the inverse of s a function? How do you know?

Other

46. Solve the equation [pic]. Explain your process for solving and justify each step. Show how to check the solution.

47. The number of bacteria present in a culture after t minutes is given as [pic]. There are 4134 bacteria present after 4 minutes. Find k. Explain how you solve this problem and justify your steps.

48. Consider the function [pic].

a. Find the domain and range of f.

b. Find [pic].

c. Find the domain and range of [pic].

d. Is [pic] a function? Explain.

CC3 Unit 5 Question Bank

Answer Section

MULTIPLE CHOICE

1. ANS: D OBJ: 8-3.1 Writing and Evaluating Logarithmic Expressions

KEY: problem solving | evaluating logarithms | logarithm

2. ANS: A OBJ: 8-3.1 Writing and Evaluating Logarithmic Expressions

KEY: logarithm | logarithmic form

3. ANS: C OBJ: 8-3.1 Writing and Evaluating Logarithmic Expressions

KEY: logarithmic form | logarithm

4. ANS: C OBJ: 8-3.1 Writing and Evaluating Logarithmic Expressions

KEY: evaluating logarithms

5. ANS: A OBJ: 8-3.1 Writing and Evaluating Logarithmic Expressions

KEY: evaluating logarithms

6. ANS: B OBJ: 8-3.1 Writing and Evaluating Logarithmic Expressions

KEY: evaluating logarithms

7. ANS: A OBJ: 8-3.1 Writing and Evaluating Logarithmic Expressions

KEY: logarithmic form | logarithm | exponential form

8. ANS: C OBJ: 8-3.2 Graphing Logarithmic Functions

KEY: graphing | logarithmic function | inverse functions

9. ANS: A OBJ: 8-3.2 Graphing Logarithmic Functions

KEY: graphing | logarithmic function | translation

10. ANS: B OBJ: 8-3.2 Graphing Logarithmic Functions

KEY: graphing | logarithmic function

11. ANS: A OBJ: 8-3.2 Graphing Logarithmic Functions

KEY: graphing | logarithmic function

12. ANS: A OBJ: 8-4.1 Using the Properties of Logarithms

KEY: properties of logarithms | logarithm | Product Property of Logarithms | Power Property of Logarithms

13. ANS: A OBJ: 8-4.1 Using the Properties of Logarithms

KEY: properties of logarithms | simplifying a logarithm | Quotient Property of Logarithms

14. ANS: D OBJ: 8-4.1 Using the Properties of Logarithms

KEY: properties of logarithms | simplifying a logarithm

15. ANS: A OBJ: 8-4.1 Using the Properties of Logarithms

KEY: properties of logarithms | expanding logarithms | Quotient Property of Logarithms

16. ANS: C OBJ: 8-4.1 Using the Properties of Logarithms

KEY: properties of logarithms | expanding logarithms | Product Property of Logarithms | Power Property of Logarithms

17. ANS: B OBJ: 8-4.1 Using the Properties of Logarithms

KEY: properties of logarithms | expanding logarithms | Power Property of Logarithms | Quotient Property of Logarithms

18. ANS: B OBJ: 8-4.1 Using the Properties of Logarithms

KEY: properties of logarithms | evaluating logarithms | Quotient Property of Logarithms

19. ANS: A OBJ: 8-5.1 Solving Exponential Equations

KEY: exponential equation

20. ANS: C OBJ: 8-5.1 Solving Exponential Equations

KEY: exponential equation

21. ANS: C OBJ: 8-5.1 Solving Exponential Equations

KEY: Change of Base Formula | exponential equation

22. ANS: A OBJ: 8-5.1 Solving Exponential Equations

KEY: graphing | exponential equation

23. ANS: B OBJ: 8-5.2 Solving Logarithmic Equations

KEY: logarithmic equation | properties of logarithms

24. ANS: A OBJ: 8-5.2 Solving Logarithmic Equations

KEY: logarithmic equation | properties of logarithms

25. ANS: B OBJ: 8-5.2 Solving Logarithmic Equations

KEY: logarithmic equation | properties of logarithms

26. ANS: A OBJ: 8-5.2 Solving Logarithmic Equations

KEY: logarithmic equation | properties of logarithms

27. ANS: B OBJ: 8-5.2 Solving Logarithmic Equations

KEY: logarithmic equation | properties of logarithms

28. ANS: D OBJ: 8-6.1 Natural Logarithms

KEY: simplifying a natural logarithm | properties of logarithms

29. ANS: A OBJ: 8-6.1 Natural Logarithms

KEY: simplifying a natural logarithm | properties of logarithms

30. ANS: D OBJ: 8-6.1 Natural Logarithms

KEY: simplifying a natural logarithm | properties of logarithms

31. ANS: A OBJ: 8-6.1 Natural Logarithms

KEY: simplifying a natural logarithm | the number e | natural logarithmic function | inverse functions

32. ANS: D OBJ: 8-6.2 Natural Logarithmic and Exponential Equations

KEY: natural logarithmic equation | properties of logarithms

33. ANS: B OBJ: 8-6.2 Natural Logarithmic and Exponential Equations

KEY: natural logarithmic equation | properties of logarithms

34. ANS: A OBJ: 8-6.2 Natural Logarithmic and Exponential Equations

KEY: natural logarithmic equation | properties of logarithms

35. ANS: D OBJ: 8-6.2 Natural Logarithmic and Exponential Equations

KEY: exponential equation | properties of logarithms

36. ANS: A OBJ: 8-6.2 Natural Logarithmic and Exponential Equations

KEY: exponential equation | properties of logarithms

37. ANS: C OBJ: 8-6.2 Natural Logarithmic and Exponential Equations

KEY: exponential equation | properties of logarithms

38. ANS: C OBJ: 7-7.1 The Inverse of a Function

KEY: inverse relations and functions | relation | graphing

39. ANS: A OBJ: 7-7.1 The Inverse of a Function KEY: inverse relations and functions |

40. ANS: B OBJ: 7-7.1 The Inverse of a Function

KEY: graphing | inverse relations and functions

41. ANS: B OBJ: 7-7.1 The Inverse of a Function

KEY: domain | inverse relations and functions | range

ESSAY

42. ANS:

[4] a. [pic]

b. In the model, the coefficient of e is 580, the original investment. The formula for continuously compounded interest uses the number e raised to the power rt, where r is the rate as a decimal, in this case 0.1, and t is the time in years.

c. To find the number of years to reach $3600, substitute 3600 into the model.

[pic]

Dividing and rounding to the nearest year, t ≈ 18. The investment will reach $3600 in about 18 years.

[3] one error in computation or incomplete explanation

[2] two errors in computation or no explanation

[1] one correct answer with no explanation

OBJ: 8-6.2 Natural Logarithmic and Exponential Equations

KEY: compounding continuously | exponential function | exponential growth | extended response | graphing | interest rates | percent | problem solving | the number e | writing in math | rubric-based question

43. ANS:

[4] Graph the function [pic] on the calculator. On the same screen, graph the line y = 142. Find the point of intersection of the population function and the line. The intersection point of the two functions is about 8.6. This means that the population of kindergarten students will be 142 about 8.6 years after the year 2000, or in 2008.

[3] one minor error in calculation or an incomplete explanation

[2] one minor error in calculation and an incomplete explanation

[1] an answer with no explanation

OBJ: 8-5.1 Solving Exponential Equations

KEY: exponential function | graphing | problem solving | rubric-based question | writing in math

44. ANS:

[4] a.

|[pic] | | |

|[pic] | |Substitute 8.4 for P. |

|[pic] | |Divide each side by 14.7. |

|[pic] | |Take the natural logarithm of each side. |

|[pic] | |Simplify. |

|[pic] | |Divide each side by –0.21. |

|[pic] | |Use a calculator. |

The elevation is about 2.7 miles above sea level.

b. The average atmospheric pressure at sea level is 14.7 lb/in.2 because x is 0 and [pic].

[3] one mathematical error or one incorrect answer

[2] two mathematical errors or one error and an incomplete explanation

[1] one correct answer with no explanation

OBJ: 8-6.2 Natural Logarithmic and Exponential Equations

KEY: exponential equation | properties of logarithms | problem solving | extended response | rubric-based question | writing in math

45. ANS:

|[4] |a. |[pic] |

| |b. |[pic] |

| |c. |If you reflect each point of s over the line y = x, you get the inverse of s. |

| |d. |Yes; it passes the vertical line test |

| |e. |No; it does not pass the vertical line test. |

|[3] |only three parts correct | |

|[2] |only two parts correct | |

|[1] |only one parts correct | |

OBJ: 7-7.1 The Inverse of a Function

KEY: extended response | relation | graphing | inverse relations and functions | rubric-based question | writing in math

OTHER

46. ANS:

To solve the logarithmic equation, you must first isolate the term with the log and then write each side in exponential form. Then solve the equation for x.

|[pic] | |

|[pic] |Subtract 5 from each side. |

|[pic] |Write in exponential form. |

|[pic] |Divide each side by 3. |

To check the solution, replace x with 0.0033.

[pic]

[pic]

[pic]

3 = 3

OBJ: 8-5.2 Solving Logarithmic Equations

KEY: logarithmic equation | writing in math | reasoning

47. ANS:

To solve this problem, substitute 4134 for B and 4 for t. Use properties of natural logarithms to solve the resulting equation. Use a calculator to find the value of k.

|[pic] |Formula |

|[pic] |Substitute 3005 for B and 6 for t. |

|[pic] |Divide each side by 1000. |

|[pic] |Take the natural logarithm of each side. |

|[pic] |Simplify. |

|[pic] |Divide each side by 6. |

|[pic] |Use a calculator. |

OBJ: 8-6.2 Natural Logarithmic and Exponential Equations

KEY: exponential equation | properties of logarithms | problem solving | writing in math

48. ANS:

a. domain of f: x ≥ –8, range of f: y ≥ 0

b. [pic]

c. domain of [pic]: all real numbers, range of f: y ≥ –8

d. [pic] is a function. For each x in the domain of [pic], there is only one value of [pic](x).

OBJ: 7-7.1 The Inverse of a Function

KEY: reasoning | domain | inverse relations and functions | multi-part question | relation | writing in math | range

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