AP Calculus AB and BC glossary

Logarithm properties

Also called: Log rules, Laws of logarithms, Log properties

The logarithm properties say the log of a product is the sum of the logs, the log of a quotient is the difference, and the log of a power moves the exponent out front as a coefficient. In calculus they are applied before differentiating, to turn one hard derivative into several easy ones.

\ln(ab) = \ln a + \ln b, \quad \ln\left(\frac{a}{b}\right) = \ln a - \ln b, \quad \ln(a^{n}) = n\ln a

The payoff is the rewrite. Differentiating $y = \ln\left(\frac{x^{2}\sqrt{x+1}}{x-3}\right)$ as written needs the chain rule wrapped around a quotient rule wrapped around a product rule. Expanded first as $2\ln x + \frac{1}{2}\ln(x+1) - \ln(x-3)$ , it is three one-step derivatives and the answer is $\frac{2}{x} + \frac{1}{2(x+1)} - \frac{1}{x-3}$ .

The same three properties are the engine of logarithmic differentiation, which takes $\ln$ of both sides precisely so the right side collapses into a sum. One domain caveat: $\ln(x^{2}) = 2\ln x$ only holds for $x > 0$ , since the left side is defined for negative $x$ and the right side is not. The honest identity is $\ln(x^{2}) = 2\ln\left|x\right|$ .

The mistake

Confusing $\ln x^{2}$ with $(\ln x)^{2}$ . The power property only moves an exponent that sits on the argument, so $\ln x^{2} = 2\ln x$ and its derivative is $\frac{2}{x}$ . The square of a logarithm needs the chain rule instead, giving $\frac{2\ln x}{x}$ .

Appears in: Unit 3: Chain Rule, Implicit, and Inverses, Unit 6: Integration and Accumulation