Precalculus · Unit 2: Exponential and Logarithmic Functions · 14 min read · Updated 2026-05-11
Logarithmic function manipulation — AP Precalculus
AP Precalculus · Unit 2: Exponential and Logarithmic Functions · 14 min read
1. Core Logarithm Rules for Expansion★★☆☆☆⏱ 4 min
All logarithm manipulation rules are direct corollaries of exponent rules, since by definition $\log_b(a) = x$ is equivalent to $b^x = a$. The product, quotient, and power rules follow directly from matching exponent properties, and are used to expand complex logarithmic expressions into simpler terms.
Exam tip: When expanding, always rewrite roots as fractional exponents before applying the power rule to avoid swapping the exponent value.
2. Condensing Logarithmic Expressions★★☆☆☆⏱ 3 min
Condensing is the reverse process of expanding: we combine multiple logarithmic terms into a single simplified logarithm. This is most often required before solving logarithmic equations, or when rewriting logarithmic functions to identify key features like intercepts or asymptotes. Always move coefficients to exponents first before combining terms to avoid misapplying rules.
Exam tip: Never add coefficients of logs with different arguments. Always move coefficients to exponents first before combining any terms.
3. Change of Base Formula★★★☆☆⏱ 3 min
The change of base formula allows us to rewrite a logarithm of any base into a ratio of logarithms with a new base of our choice. On the AP exam, this is used to evaluate non-standard base logarithms with a calculator, or to convert all terms in an expression to the same base for further manipulation.
Exam tip: Always double-check the order of numerator and denominator: original argument goes in the numerator, original base goes in the denominator. Swapping gives the reciprocal of the correct answer.
4. AP-Style Worked Examples★★★☆☆⏱ 4 min
Common Pitfalls
Why: Students confuse the product rule for $\log_b(mn)$ with a sum inside the logarithm. No general rule exists for sums of arguments.
Why: Students confuse the change of base ratio of two logs with the quotient rule for a division inside a single log.
Why: $\ln x^2$ is defined for all $x \neq 0$, but $2\ln x$ is only defined for $x>0$, so domains do not match.
Why: Students misinterpret the power rule, confusing the exponent on the argument with an exponent on the entire logarithm.
Why: Students forget that all core logarithm combination rules require terms to have the same base.