AP®MathsCollege BoardPrecalculusExam QuestionsExponential & Logarithmic FunctionsLogarithmic Functions

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

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Logarithmic Functions

1
1 mark

The function space j is given by space j open parentheses x close parentheses equals log subscript 2 x plus 3 log subscript 2 2.

Rewrite space j open parentheses x close parentheses as a single logarithm base 2 without negative exponents in any part of the expression. Your result should be of the form log subscript 2 open parentheses expression close parentheses.

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2
1 mark

The function space j is given by space j open parentheses x close parentheses equals log subscript 10 open parentheses 8 x to the power of 5 close parentheses plus log subscript 10 open parentheses 2 x squared close parentheses minus 9 log subscript 10 x.

Rewrite space j open parentheses x close parentheses as a single logarithm base 10 without negative exponents in any part of the expression. Your result should be of the form log subscript 10 open parentheses expression close parentheses.

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3
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1 mark

The function space f is given by space f open parentheses x close parentheses equals ln x.

Solve space f open parentheses x close parentheses equals 9 for values of x in the domain of space f.

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4
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1 mark

The function g is given by g open parentheses x close parentheses equals log subscript 3 open parentheses 3 to the power of x close parentheses.

Rewrite g open parentheses x close parentheses as an expression that does not involve a logarithm.

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5
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1 mark

The function h is given by h open parentheses x close parentheses equals e to the power of x ln 4 end exponent.

Rewrite h open parentheses x close parentheses as an equivalent expression of the form h open parentheses x close parentheses equals b to the power of x, where b is a constant. Identify the value of b.

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6
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1 mark

The function space j is given byspace j open parentheses x close parentheses equals 3 log subscript 5 open parentheses 2 x close parentheses minus 1 third log subscript 5 x

Rewrite space j open parentheses x close parentheses as a single logarithm base 5 without negative exponents in any part of the expression. Your result should be of the form log subscript 5 open parentheses expression close parentheses.

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7
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1 mark

The function g is given by g open parentheses x close parentheses equals 2 ln x plus 1 third ln x.

Rewrite g open parentheses x close parentheses as a single logarithm without negative exponents in any part of the expression. Your result should be in the form ln open parentheses expression close parentheses.

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8
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2 marks

The function space f is given by space f open parentheses x close parentheses equals 6 to the power of x.

Rewrite space f open parentheses x close parentheses in the equivalent form space f open parentheses x close parentheses equals e to the power of k x end exponent. Express the exact value of k using a natural logarithm, then express k as a decimal approximation.

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9
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2 marks

The function g is given by g open parentheses x close parentheses equals negative 3 log subscript 2 open parentheses x minus 4 close parentheses plus 1.

State whether the function g is increasing or decreasing on its domain. Then determine the end behavior of g as x increases without bound. Express your answer using the mathematical notation of a limit.

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10
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2 marks

On a base-10 logarithmic scale measuring earthquake magnitudes, Earthquake A has a measurement of 4.5 and Earthquake B has a measurement of 6.2.

Determine how many times greater the actual amplitude of Earthquake B is than the actual amplitude of Earthquake A. Express your answer as a decimal approximation. Show the work that leads to your answer.

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11
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1 mark

The functions space f and g are given by space f open parentheses x close parentheses equals 5 to the power of x and g open parentheses x close parentheses equals log subscript 5 x.

Determine the exact value of f open parentheses g open parentheses 14 close parentheses close parentheses.

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12
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1 mark

Determine the exact value of log subscript 4 open parentheses 1 over 8 close parentheses. Show the work that leads to your answer.

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13
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2 marks

The function h is given by h open parentheses x close parentheses equals log subscript 7 x.

Rewrite h open parentheses 20 close parentheses using the change of base property with the natural logarithm. Then evaluate the expression as a decimal approximation.

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14
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1 mark

The function space j is given by space j open parentheses x close parentheses equals log subscript 2 open parentheses x to the power of 4 close parentheses minus 3 log subscript 2 open parentheses square root of x close parentheses plus 2 log subscript 2 open parentheses 8 x close parentheses, defined for x > 0.

Rewrite space j open parentheses x close parentheses as a single logarithm base 2 without negative exponents in any part of the expression. Your result should be of the form log subscript 2 open parentheses c x to the power of n close parentheses, where c and n are constants.

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15
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1 mark

The function space f is given by space f open parentheses x close parentheses equals ln open parentheses e to the power of 2 x plus 1 end exponent close parentheses plus ln open parentheses e to the power of x minus 3 end exponent close parentheses minus 2 ln open parentheses square root of e to the power of x end root close parentheses, defined for all real numbers.

Rewrite space f open parentheses x close parentheses as a single linear expression in x.

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16a
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1 mark

The function g is given by g open parentheses x close parentheses equals 2 log subscript 5 open parentheses 3 x plus 1 close parentheses minus 4, defined for x greater than negative 1 third.

Let g to the power of negative 1 end exponent be the inverse function of g. Find the value of g to the power of negative 1 end exponent open parentheses 2 close parentheses as a decimal approximation, or indicate that it is not defined.

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16b
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1 mark

Find all values of x for which g open parentheses x close parentheses equals 0, as a decimal approximation, or indicate there are no such values.

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