College Board · AP Precalculus

AP Precalculus: Exponential and Logarithmic Functions — Practice Questions & Answers

Q: What is the common ratio of the geometric sequence $3, 6, 12, 24, \ldots$?

$2$. The common ratio is found by dividing consecutive terms: $\frac{6}{3}=2$.

Q: Which function represents exponential growth?

$f(x)=3(2)^x$. For $f(x)=ab^x$ with $a>0$, growth occurs when the base $b>1$. Here $b=2>1$.

Q: What is the value of $\log_2 8$?

$3$. $\log_2 8$ asks the power of $2$ that gives $8$. Since $2^3=8$, the value is $3$.

Q: Which expression equals $\log_b(xy)$?

$\log_b x+\log_b y$. The product property of logarithms states $\log_b(xy)=\log_b x+\log_b y$.

Q: The function $g(x)=\log_3 x$ is the inverse of which function?

$f(x)=3^x$. The logarithm base $3$ is the inverse of the exponential function with base $3$, namely $f(x)=3^x$.

Q: Which of the following is the vertical asymptote of $f(x)=\log_4 x$?

$x=0$. Logarithmic functions of the form $\log_b x$ have a vertical asymptote at $x=0$.

Q: A quantity changes by equal differences over equal-length input intervals. Which model best describes it?

Linear. Equal differences (constant rate of change) over equal intervals is the defining feature of a linear model.

Q: What is the $10$th term of the arithmetic sequence with first term $a_1=4$ and common difference $d=3$?

$31$. Using $a_n=a_1+(n-1)d$, we get $a_{10}=4+(10-1)(3)=4+27=31$.

Q: What is the $6$th term of the geometric sequence with $a_1=5$ and common ratio $r=2$?

$160$. Using $a_n=a_1 r^{n-1}$, we get $a_6=5(2)^{5}=5(32)=160$.

Q: An exponential function $f$ has $f(0)=8$ and a constant proportional change such that $f(1)=12$. What is $f(2)$?

$18$. The ratio is $\frac{12}{8}=1.5$, so $f(2)=12(1.5)=18$.

Unit 2 builds from arithmetic and geometric sequences into exponential functions defined by constant proportional change, then introduces logarithms as their inverses. You will model growth and decay, compare polynomial, exponential, and logarithmic behavior, apply logarithm properties, solve exponential and logarithmic equations, work with inverse and composite functions, and linearize data using semi-log plots.

162 practice questions available for this topic — here are 10 with full answers and explanations.

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Practice questions with answers

1 Multiple choice · Easy

What is the common ratio of the geometric sequence $3, 6, 12, 24, \ldots$?

$2$
$3$
$\frac{1}{2}$
$9$

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2 Multiple choice · Easy

Which function represents exponential growth?

$f(x)=3(2)^x$
$f(x)=3(0.5)^x$
$f(x)=3x+2$
$f(x)=3(0.9)^x$

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3 Multiple choice · Easy

What is the value of $\log_2 8$?

$3$
$4$
$2$
$16$

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4 Multiple choice · Easy

Which expression equals $\log_b(xy)$?

$\log_b x+\log_b y$
$\log_b x \cdot \log_b y$
$\log_b x-\log_b y$
$\frac{\log_b x}{\log_b y}$

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5 Multiple choice · Easy

The function $g(x)=\log_3 x$ is the inverse of which function?

$f(x)=3^x$
$f(x)=x^3$
$f(x)=3x$
$f(x)=\frac{1}{3}x$

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6 Multiple choice · Easy

Which of the following is the vertical asymptote of $f(x)=\log_4 x$?

$x=0$
$y=0$
$x=1$
$x=4$

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7 Multiple choice · Easy

A quantity changes by equal differences over equal-length input intervals. Which model best describes it?

Linear
Exponential
Quadratic
Logarithmic

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8 Multiple choice · Medium

What is the $10$th term of the arithmetic sequence with first term $a_1=4$ and common difference $d=3$?

$31$
$34$
$30$
$27$

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9 Multiple choice · Medium

What is the $6$th term of the geometric sequence with $a_1=5$ and common ratio $r=2$?

$160$
$320$
$60$
$64$

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10 Multiple choice · Medium

An exponential function $f$ has $f(0)=8$ and a constant proportional change such that $f(1)=12$. What is $f(2)$?

$18$
$16$
$24$
$20$

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Key terms in Exponential and Logarithmic Functions

Arithmetic Sequence: A sequence in which each term differs from the previous by a constant common difference.

Geometric Sequence: A sequence in which each term is the previous term multiplied by a constant common ratio.

Common Difference: The constant amount added between consecutive terms of an arithmetic sequence.

Common Ratio: The constant factor multiplied between consecutive terms of a geometric sequence.

Sequence: An ordered list of numbers following a particular rule or pattern.

Term of a Sequence: An individual number in a sequence, often indexed by position $n$.

Index: The position number of a term within a sequence.

Explicit Formula: A formula expressing the nth term of a sequence directly in terms of $n$.

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AP Precalculus: Exponential and Logarithmic Functions — Practice Questions & Answers

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Key terms in Exponential and Logarithmic Functions

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