AP Calculus AB and BC glossary

Natural logarithm

The natural logarithm is the logarithm to base e. Its derivative is one over x, and reading that backwards makes it the antiderivative of one over x, which is the single case the power rule cannot handle.

\frac{d}{dx}\ln x = \frac{1}{x}, \quad \int \frac{1}{x}\,dx = \ln\left|x\right| + C

The absolute value in the antiderivative matters. The domain of $\ln x$ is positive $x$ , but $\frac{1}{x}$ is defined for negative $x$ too, and the bars extend the antiderivative to cover that.

The mistake

Splitting the logarithm of a sum. There is no rule for the logarithm of x plus y; the product and quotient rules apply to products and quotients only.

Appears in: Unit 2: Defining the Derivative, Unit 6: Integration and Accumulation