$ \begin{array}{c@{\qquad}c} \textbf{Logarithm Identities} & \textbf{Natural Log Identities} \\[8pt] \log(xy) = \log(x) + \log(y) & \ln(xy) = \ln(x) + \ln(y) \\[6pt] \log\!\left(\frac{x}{y}\right) = \log(x) - \log(y) & \ln\!\left(\frac{x}{y}\right) = \ln(x) - \ln(y) \\[6pt] \log(x^a) = a\log(x) & \ln(x^a) = a\ln(x) \\[6pt] \log(10^x) = x & \ln(e^x) = x \\[6pt] 10^{\log(x)} = x & e^{\ln(x)} = x \end{array} $

Derivatives

$\frac{d}{dx} \log_a(x) = \frac{1}{x \ln(a)}$

$\frac{d}{dx} \log_a(f(x)) = \frac{f'(x)}{f(x)\ln(a)}$

$\frac{d}{dx} \ln(x) = \frac{1}{x}$

$\frac{d}{dx} \ln|x| = \frac{1}{x}$

$\frac{d}{dx} \ln(f(x)) = \frac{f'(x)}{f(x)}$

Integrals

$\int \frac{1}{x} \, dx = \ln|x| + C$

$\int \frac{f'(x)}{f(x)} \, dx = \ln|f(x)| + C$