AP Exam Prep · AP Precalculus

AP Precalculus Unit 2 Practice Test: Exponential and Logarithmic Functions

Unit 2 contrasts additive and multiplicative change: arithmetic sequences and linear functions against geometric sequences and exponential functions. It then builds composition and inverses, which lead to logarithms, log laws, solving exponential and log equations, and modelling with semi-log plots.

Exam weighting:
25–40%
Questions:
23 multiple choice
Time:
about 58 minutes
Cost:
Free, no sign-up

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Question 1

Topic 2.1 · Apply it

An arithmetic sequence has a3 = 11 and a7 = 27. What is a20?

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What AP Precalc Unit 2 covers

Unit 2, Exponential and Logarithmic Functions, makes up 25–40% of the multiple-choice section and usually takes ~30–45 class periods. These are the topics in the current course framework, and every question in the test above is tagged with one of them.

  1. 2.1Change in Arithmetic and Geometric Sequences
  2. 2.2Change in Linear and Exponential Functions
  3. 2.3Exponential Functions
  4. 2.4Exponential Function Manipulation
  5. 2.5Exponential Function Context and Data Modeling
  6. 2.6Competing Function Model Validation
  7. 2.7Composition of Functions
  8. 2.8Inverse Functions
  9. 2.9Logarithmic Expressions
  10. 2.10Inverses of Exponential Functions
  11. 2.11Logarithmic Functions
  12. 2.12Logarithmic Function Manipulation
  13. 2.13Exponential and Logarithmic Equations and Inequalities
  14. 2.14Logarithmic Function Context and Data Modeling
  15. 2.15Semi-log Plots

Unit 2 key ideas

Linear adds, exponential multiplies

Over equal input steps a linear function adds the same amount and an exponential function multiplies by the same ratio. In f(x) = a·bˣ, a is the initial value and b is the growth (b > 1) or decay (0 < b < 1) factor; growth of r% per step means b = 1 + r.

Inverses undo, and logs are inverses of exponentials

If f(a) = b then f⁻¹(b) = a, and the graph of an inverse is the reflection in y = x. log_b(x) = y means bʸ = x, so y = log_b(x) is the inverse of y = bˣ: domain x > 0, a vertical asymptote at x = 0, and the point (1, 0).

Log laws turn products into sums

log_b(MN) = log_b M + log_b N, log_b(M/N) = log_b M − log_b N and log_b(Mᵖ) = p·log_b M. Change of base: log_b x = ln x ÷ ln b. Solving bˣ = c means x = log_b c.

Choose and check models

Equal differences suggest a linear model, equal ratios an exponential one. A residual plot with no pattern supports a model. On a semi-log plot (log scale on y), exponential data look linear.

Common Unit 2 mistakes

  • Writing log(M + N) = log M + log N; the log of a sum does not split.
  • Confusing f⁻¹(x) (the inverse function) with 1/f(x) (the reciprocal).
  • Using the percentage itself as the base: 5% growth is b = 1.05, not b = 5 or b = 0.05.

How to study this unit

  • For a table of values, check differences and ratios over equal steps before choosing a model.
  • When solving log equations, check every answer in the original equation: logs need positive inputs.

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More AP Precalculus practice tests

← Unit 1: Polynomial and Rational FunctionsUnit 3: Trigonometric and Polar Functions →

About these questions

StudyCoach wrote these questions for this test. Each one is mapped to a topic in the College Board's current AP Precalculus course framework. Every answer key was also checked by a separate AI review, and any disagreement was rechecked by hand. They have not yet been reviewed by an AP Precalculus teacher. If you think an answer is wrong, email support@studycoach.io.

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