AP®MathsCollege BoardPrecalculusExam QuestionsExponential & Logarithmic FunctionsLogarithmic Functions

Logarithmic Functions (College Board AP® Precalculus): Exam Questions

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Logarithmic Functions

1
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1 mark

Which of the following is the best approximation for the value of log subscript 7 40?

  • 0 . 548

  • 1 . 896

  • 3 . 689

  • 5 . 714

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2
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1 mark

What is the exact value of log subscript 2 32?

  • 1 fifth

  • 5

  • 16

  • 64

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3
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1 mark

What is the exact value of ln open parentheses e to the power of 4 close parentheses?

  • 1

  • e

  • 4

  • 4 e

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4
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1 mark

For x greater than 0, which of the following is equivalent to e to the power of ln open parentheses 5 x close parentheses end exponent?

  • x

  • 5 x

  • 5 e^{x}

  • e^{5 x}

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5
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1 mark

The graph of the exponential function g open parentheses x close parentheses equals b to the power of x, where b greater than 0 and b not equal to 1, passes through the point open parentheses 3 comma 8 close parentheses. Which of the following points must lie on the graph of the logarithmic function space f open parentheses x close parentheses equals log subscript b x?

  • open parentheses negative 3 comma negative 8 close parentheses

  • open parentheses 1 third comma 1 over 8 close parentheses

  • open parentheses 3 comma 8 close parentheses

  • open parentheses 8 comma 3 close parentheses

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6
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1 mark

Let m, n, and p be positive constants. Which of the following is equivalent to \log_{10}\left(\frac{m^3}{np^2}\right)?

  • \log_{10}(m + 3) - \log_{10} n - 2\log_{10} p

  • 3\log_{10} m + \log_{10} n - 2\log_{10} p

  • 3\log_{10} m - \log_{10} n - 2\log_{10} p

  • 3\log_{10} m - \log_{10} n + 2\log_{10} p

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7
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1 mark

Let a and b be positive constants. Which of the following is equivalent to 3 \ln a + 2 \ln b?

  • \ln \left(\frac{a^{3}}{b^{2}}\right)

  • \ln \left(a^{3} b^{2}\right)

  • \ln \left(3 a + 2 b\right)

  • \ln \left(\frac{3 a}{2 b}\right)

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8
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1 mark

The function space f is given by space f open parentheses x close parentheses equals log subscript 3 open parentheses x minus 4 close parentheses plus 2. Which of the following describes the domain and the vertical asymptote of space f?

  • Domain: x greater than negative 4; Vertical asymptote: x equals negative 4

  • Domain: x greater than 2; Vertical asymptote: x equals 2

  • Domain: x greater than 4; Vertical asymptote: x equals 4

  • Domain: all real numbers; Vertical asymptote: space y equals 2

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9
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1 mark

On a base-10 logarithmic scale, the measurement of quantity X is 3.2 and the measurement of quantity Y is 5.6. Which of the following is the best approximation for how many times greater the actual value of quantity Y is than the actual value of quantity X?

  • 1 . 750

  • 2 . 400

  • 24 . 000

  • 251 . 189

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10
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1 mark

Let x and y be positive constants. Which of the following is equivalent to log subscript 2 open parentheses x to the power of 4 over y cubed close parentheses?

  • 4 log subscript 2 x plus 3 log subscript 2 y

  • 4 log subscript 2 x minus 3 log subscript 2 y

  • fraction numerator 4 log subscript 2 x over denominator 3 log subscript 2 y end fraction

  • open parentheses log subscript 2 x close parentheses to the power of 4 minus open parentheses log subscript 2 y close parentheses cubed

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11
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1 mark

The exponential function space f is given by space f open parentheses x close parentheses equals 5 to the power of x. The function space f can be rewritten in the equivalent form space f open parentheses x close parentheses equals e to the power of k x end exponent, where k is a constant. What is the value of k, expressed as a decimal approximation?

  • 0 . 699

  • 1 . 609

  • 5 . 000

  • 148 . 413

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12
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1 mark

The function space f is given by space f open parentheses x close parentheses equals log subscript 3 open parentheses x minus h close parentheses plus k, where h and k are constants. The graph of space f has an x-intercept at x equals 4 and a vertical asymptote at x equals 1.

What is the value of space f open parentheses 28 close parentheses?

  • 1

  • 2

  • 3

  • 4

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13
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1 mark

The decibel scale measures sound intensity by the formula L equals 10 log subscript 10 open parentheses I over I subscript 0 close parentheses, where I is the sound intensity, I subscript 0 is a reference intensity, and L is the sound level in decibels.

Sound A has a sound level of 60 decibels. Sound B has an intensity that is 1000 times greater than the intensity of sound A.

What is the sound level of sound B in decibels?

  • 30

  • 63

  • 90

  • 180

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14
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1 mark

The pH scale measures the acidity of a solution by the formula text pH end text equals negative log subscript 10 open square brackets H to the power of plus close square brackets, where open square brackets H to the power of plus close square brackets is the hydrogen ion concentration in moles per liter.

Solution X has a pH of 4.5. Solution Y has a hydrogen ion concentration that is 100 times the hydrogen ion concentration of solution X.

What is the pH of solution Y?

  • 0 . 045

  • 2 . 500

  • 4 . 500

  • 6 . 500

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15
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1 mark

If x greater than 1, which of the following is equivalent to 2 log subscript 2 open parentheses x close parentheses minus 1 half log subscript 2 open parentheses x squared minus 1 close parentheses plus log subscript 2 open parentheses 3 close parentheses?

  • \log_{2} \left(\dfrac{3 x^{2}}{x^{2} - 1}\right)

  • \log_{2} \left(\dfrac{3 x^{2}}{\sqrt{x^{2} - 1}}\right)

  • \log_{2} \left(\dfrac{6 x}{\sqrt{x^{2} - 1}}\right)

  • \log_{2} \left(\dfrac{x^{2} \sqrt{x^{2} - 1}}{3}\right)

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16
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1 mark

Consider the function space f defined by space f open parentheses x close parentheses equals negative log subscript 3 open parentheses x minus 2 close parentheses plus 1. Which of the following statements about space f is true?

  • space f is decreasing on open parentheses 2 comma infinity close parentheses, and space f open parentheses x close parentheses rightwards arrow negative infinity as x rightwards arrow 2 to the power of plus.

  • space f is decreasing on open parentheses 2 comma infinity close parentheses, and space f open parentheses x close parentheses rightwards arrow infinity as x rightwards arrow 2 to the power of plus.

  • space f is increasing on open parentheses 2 comma infinity close parentheses, and space f open parentheses x close parentheses rightwards arrow negative infinity as x rightwards arrow infinity.

  • space f is increasing on open parentheses 2 comma infinity close parentheses, and space f open parentheses x close parentheses rightwards arrow infinity as x rightwards arrow infinity.

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