Precalculus · Unit 2: Exponential and Logarithmic Functions · 14 min read · Updated 2026-05-11
Logarithmic expressions — AP Precalculus
AP Precalculus · Unit 2: Exponential and Logarithmic Functions · 14 min read
1. Definition and Logarithm-Exponential Equivalence★★☆☆☆⏱ 3 min
A logarithmic expression is any algebraic expression containing one or more logarithms, which are the inverse functions of exponential functions. Logarithms answer the question: what exponent do I raise a given base to, to get the argument value? On the AP exam, you will primarily work with common logarithms (base 10, written $ ext{log }x$) and natural logarithms (base $e$, written $ ext{ln }x$), though any positive base $b \neq 1$ is valid. Mastery of this topic is required for nearly all subsequent Unit 2 content, so errors here cascade into higher-weight problems.
Exam tip: If you are ever unsure of a logarithm value, convert it back to exponential form to check—this takes 10 seconds and eliminates almost all sign errors.
2. Core Logarithm Properties: Expanding and Condensing★★★☆☆⏱ 4 min
Because logarithms are inverses of exponents, their properties directly correspond to familiar exponent rules. For any positive base $b \neq 1$, positive arguments $M, N$, and real number $k$, the three core properties are used for two common AP exam tasks: expanding a single condensed expression into simpler terms, or condensing a sum/difference of logs into a single expression.
Product Rule: $\log_b(MN) = \log_b M + \log_b N$ (corresponds to $b^m b^n = b^{m+n}$)
Quotient Rule: $\log_b\left(\frac{M}{N}\right) = \log_b M - \log_b N$ (corresponds to $\frac{b^m}{b^n} = b^{m-n}$)
Power Rule: $\log_b(M^k) = k \log_b M$ (corresponds to $(b^m)^k = b^{mk}$)
Exam tip: When condensing expressions, always move coefficients inside logarithms as exponents before combining terms with product/quotient rules—this avoids common coefficient errors.
3. Change of Base Formula★★★☆☆⏱ 3 min
The change of base formula rewrites a logarithm of any base as a ratio of logarithms with another base. This is required to evaluate any non-common/non-natural logarithm with a calculator, and to simplify expressions with mixed bases.
Exam tip: When asked to evaluate a logarithm where both the base and argument are powers of a common smaller base, always rewrite to cancel logs and avoid unnecessary calculator work.
4. AP-Style Worked Practice★★★★☆⏱ 4 min
Common Pitfalls
Why: Students confuse the product rule for logs with addition inside the log; no general rule exists for the logarithm of a sum.
Why: Students confuse the change of base formula (quotient of two separate logs) with the quotient rule (which applies to a quotient inside a single log).
Why: Students incorrectly apply the power rule to a negative argument, forgetting the domain requirement that all logarithm arguments must be positive.
Why: Students misapply the power rule, which only applies to a power inside the logarithm, not to the entire logarithm raised to a power.
Why: Students incorrectly apply reciprocal rules from general algebra to logarithms, forgetting the power rule for reciprocals.