AP Precalculus, AP exams, 25–40%

Unit 2: Exponential and Logarithmic Functions

Unit 2 develops proportional growth. Arithmetic and geometric sequences lead into linear and exponential models, composition and inverses lead into logarithms, and the unit closes with data modeling and semi-log plots. Recognizing whether a data set is better modeled by a linear, quadratic, or exponential function is an explicitly tested judgment.

Position: 2 of 4 units
Published weighting: 25–40% in the official framework
Topics covered: 6 topics in this guide

What is covered

Topics inside this unit.

These topics describe the scope students most often review for this part of AP Precalculus. Treat them as a study checklist rather than an exhaustive list of what a single exam form can sample.

Sequences and their function models: Arithmetic and geometric sequences and the linear and exponential functions that extend them.
Exponential functions: Growth and decay behavior, the effect of the base, and manipulating exponential expressions.
Composition and inverse functions: Building composite functions, finding inverses, and the domain restrictions inverses require.
Logarithmic functions: Logarithms as inverses of exponentials, logarithm properties, and graph behavior including asymptotes.
Exponential and logarithmic equations: Solving equations and inequalities, including extraneous solutions produced by domain restrictions.
Model selection and semi-log plots: Choosing among competing models for a data set, and reading a semi-log plot as evidence of exponential behavior.

Study tips

How to practice Exponential and Logarithmic.

Section-specific guidance that follows from how this material is usually tested.

Practice logarithm properties in both directions, expanding and condensing, until either is immediate.
Check the domain after solving any logarithmic equation; extraneous solutions are a standard test.
Learn to recognize an exponential model from a table by constant ratios rather than constant differences.

Where this fits

Unit 2: Exponential and Logarithmic Functions is 2 of 4 units in the AP Precalculus guide, and it carries a published weighting of 25–40% in the official framework. Use the full guide to balance it against everything else you have to review.

Back to the AP Precalculus guide Official exam information

Sample Exponential and Logarithmic questions

Practice questions, not yet expert-reviewed

Original practice questions written to the published framework for this section. No expert has reviewed them, and they are not taken from any official exam. Try each one before you open the answer.

Question 1Foundational
What is the seventh term of the geometric sequence 3, 6, 12, 24, ...?
1. A192
2. B96
3. C384
4. D129
Show answer
Answer: A. 192
The sequence has first term 3 and common ratio 2, so the nth term is 3 times 2^(n - 1). For n = 7 that is 3 times 2^6 = 3(64) = 192. Answering 96 uses 2^5 and stops one term early, an off-by-one error in the exponent.
Key insight: In a geometric sequence the exponent is n - 1, because the first term uses ratio to the zero power.
Question 2Foundational
If log base 2 of x equals 5, what is x?
1. A32
2. B25
3. C10
4. D7
Show answer
Answer: A. 32
The logarithmic equation says 2^5 = x, so x = 32. Answering 25 squares the base and exponent in the wrong order, and 10 confuses the base 2 with the common logarithm base.
Key insight: A logarithm is an exponent: log base b of x = y means b^y = x.
Question 3Intermediate
What is the solution of the equation 3^(x + 1) = 243?
1. Ax = 4
2. Bx = 5
3. Cx = 3
4. Dx = 81
Show answer
Answer: A. x = 4
Writing 243 as 3^5 makes the equation 3^(x + 1) = 3^5, so x + 1 = 5 and x = 4. Answering 5 solves for the exponent x + 1 and forgets the final subtraction.
Key insight: Rewrite both sides with the same base, then set the exponents equal.
Question 4Intermediate
Is the following statement true or false: for all positive numbers a and b, log(a + b) equals log(a) + log(b)?
1. ATrue
2. BFalse
Show answer
Answer: B. False
The product rule says log(ab) = log(a) + log(b); addition inside the logarithm does not distribute. Testing a = b = 1 shows it: log(2) is about 0.301 while log(1) + log(1) = 0. There is no logarithm rule that simplifies log(a + b).
Key insight: Logarithms convert multiplication into addition, never addition into addition.
Question 5Advanced
A scatterplot of ln(y) against x for a data set is closely linear with positive slope. Which model best fits the original data?
1. AAn exponential model
2. BA linear model
3. CA quadratic model
4. DA logarithmic model
Show answer
Answer: A. An exponential model
If ln(y) = mx + b, then exponentiating gives y = e^b times e^(mx), which is exponential in x. A semi-log plot that straightens the data is therefore evidence of exponential behavior. A linear model would already look straight on ordinary axes, with no transformation needed.
Key insight: Linearity on a semi-log plot of ln(y) versus x is the signature of an exponential model.

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Unit 2: Exponential and Logarithmic Functions