AP®MathsCollege BoardPrecalculusExam QuestionsExponential & Logarithmic FunctionsExponential & Logarithmic Equations & Inequalities

Exponential & Logarithmic Equations & Inequalities (College Board AP® Precalculus): Exam Questions

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Exponential & Logarithmic Equations & Inequalities

1
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1 mark

The function g is given by g open parentheses x close parentheses equals ln open parentheses x plus 2 close parentheses minus 1. Which of the following is an expression for g to the power of negative 1 end exponent open parentheses x close parentheses?

  • e to the power of x minus 2 end exponent plus 1

  • e to the power of x minus 1 end exponent plus 2

  • e to the power of x plus 1 end exponent minus 2

  • e to the power of x plus 2 end exponent minus 1

Choose your answer

2
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1 mark

What are all values of x for which 9 to the power of x plus 2 end exponent equals 27 to the power of x?

  • x equals negative 2

  • x equals 1

  • x equals 2

  • x equals 4

Choose your answer

3
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1 mark

What is the value of x for which log subscript 2 x equals 5?

  • 10

  • 25

  • 32

  • 64

Choose your answer

4
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1 mark

The function space f is given by space f open parentheses x close parentheses equals e to the power of x minus 4.

Which of the following is an expression for space f to the power of negative 1 end exponent open parentheses x close parentheses?

  • \ln \left(x - 4\right)

  • \ln \left(x + 4\right)

  • \ln x - 4

  • \ln x + 4

Choose your answer

5
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1 mark

The function P is given by P open parentheses t close parentheses equals fraction numerator 800 comma 000 over denominator 1 plus 0.6 e to the power of k t end exponent end fraction, where k is a constant. If P open parentheses 5 close parentheses equals 450 comma 000, what is the value of P \left(15\right) ?

  • 240 comma 000

  • 346 comma 776

  • 398 comma 361

  • 612 comma 105

Choose your answer

6
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1 mark

The function space f is given by space f open parentheses x close parentheses equals 4 to the power of x minus 3 end exponent plus 5. Which of the following is an expression for space f to the power of negative 1 end exponent open parentheses x close parentheses?

  • log subscript 4 open parentheses x minus 5 close parentheses plus 3

  • log subscript 4 open parentheses x minus 3 close parentheses plus 5

  • log subscript 4 open parentheses x plus 3 close parentheses minus 5

  • log subscript 4 open parentheses x plus 5 close parentheses minus 3

Choose your answer

7
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1 mark

The function P is given by P \left(t\right) = \frac{500 , 000}{1 + 0 . 5 e^{k t}}, where k is a constant. If P \left(6\right) = 320 , 000, what is the value of P \left(18\right) ?

  • 186 , 047

  • 292 , 071

  • 306 , 220

  • 702 , 331

Choose your answer

8
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1 mark

What are all values of x for which \ln(x^5) - \ln x = 8?

  • x = -e^2 \textrm{ and } x = e^2

  • x = e only

  • x = e^2 only

  • x = e^8 only

Choose your answer

9
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1 mark

What are all values of x for which log subscript 2 x plus log subscript 2 open parentheses x minus 3 close parentheses equals 2?

  • x equals negative 1 only

  • x equals negative 1 and x equals 4

  • x equals 4 only

  • x equals 4 and x equals 7

Choose your answer

10
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1 mark

The function h is given by h open parentheses x close parentheses equals 5 times open parentheses 1.4 close parentheses to the power of x. What are all values of x for which h open parentheses x close parentheses greater or equal than 15?

  • x less or equal than 3.265

  • x greater or equal than 1.099

  • x greater or equal than 3.265

  • x greater or equal than 10.714

Choose your answer

11
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1 mark

What are all values of x for which 2 ln open parentheses x plus 1 close parentheses less than 4?

  • x less than e squared minus 1

  • negative 1 less than x less than e squared minus 1

  • 0 less than x less than e squared minus 1

  • x greater than e squared minus 1

Choose your answer

12
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1 mark

The temperature of a liquid, in degrees Celsius, t minutes after being placed in a freezer is modeled by the function T open parentheses t close parentheses equals 15 times open parentheses 0.8 close parentheses to the power of t minus 5. What are all values of t for which T open parentheses t close parentheses equals negative 10, or indicate that there are no such values?

  • t equals negative 1.246

  • t equals 0.528

  • t equals 4.923

  • There are no such values of t

Choose your answer

13
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1 mark

Consider the functions f and g given by f open parentheses x close parentheses equals log subscript 3 open parentheses x minus 2 close parentheses plus log subscript 3 open parentheses x plus 4 close parentheses and g open parentheses x close parentheses equals log subscript 3 open parentheses x plus 10 close parentheses. In the x y-plane, what are all x-coordinates of the points of intersection of the graphs of f and g?

  • x equals fraction numerator negative 1 plus square root of 73 over denominator 2 end fraction only

  • x = 6

  • x equals fraction numerator negative 1 minus square root of 73 over denominator 2 end fraction and x equals fraction numerator negative 1 plus square root of 73 over denominator 2 end fraction

  • x = - 6 and x = - 3

Choose your answer

14
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1 mark

The function space f is given by space f open parentheses x close parentheses equals negative 3 e to the power of x minus 2 end exponent plus 4. Let space f to the power of negative 1 end exponent be the inverse function of space f. What is the domain of space f to the power of negative 1 end exponent?

  • open parentheses negative infinity comma negative 2 close parentheses

  • open parentheses negative infinity comma 4 close parentheses

  • open parentheses negative infinity comma infinity close parentheses

  • open parentheses 4 comma infinity close parentheses

Choose your answer

15
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1 mark

What are all values of x for which e to the power of 2 x end exponent minus 9 e to the power of x plus 14 equals 0?

  • x equals ln 2 only

  • x equals ln 7 only

  • x equals ln 14 only

  • x equals ln 2 and x equals ln 7

Choose your answer

16
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1 mark

The function g is given by g open parentheses x close parentheses equals 4 minus 2 log subscript 3 open parentheses x plus 5 close parentheses.

Which of the following is an expression for g to the power of negative 1 end exponent open parentheses x close parentheses?

  • 3^{\frac{x - 4}{2}} - 5

  • 3^{\frac{4 - x}{2}} - 5

  • 3^{\frac{4 - x}{2}} + 5

  • 3^{4 - x} - 5

Choose your answer

17
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1 mark

Find all real solutions to e to the power of 2 x end exponent minus 5 e to the power of x plus 6 equals 0.

  • x equals 2 or x equals 3

  • x equals ln 2 or x equals ln 3

  • x equals ln 5

  • x equals ln 6

Choose your answer

18
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1 mark

How many real values of x satisfy the equation log subscript 2 open parentheses x close parentheses plus log subscript 2 open parentheses x minus 3 close parentheses equals 2?

  • 0

  • 1

  • 2

  • 4

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