Logarithm Rules

Product

$\log_b(xy) = \log_b x + \log_b y$

Quotient

$\log_b(x/y) = \log_b x - \log_b y$

Power

$\log_b(x^p) = p \, \log_b x$

Change of base

$\log_b x = \dfrac{\log_k x}{\log_k b}$

Log of base

$\log_b b = 1$

Log of one

$\log_b 1 = 0$

Inverse of exponent

$\log_b(b^x) = x, \quad b^{\log_b x} = x$

Definition

$\ln x = \log_e x$

Inverse pair

$\ln(e^x) = x, \quad e^{\ln x} = x$

Special values

$\ln 1 = 0, \quad \ln e = 1$

Exponent to base e

$a^x = e^{x \ln a}$

Log-sum-exp identity

$\ln \sum_i e^{z_i}$ is a smooth max used in softmax stability

Derivative

$\dfrac{d}{dx} \ln x = \dfrac{1}{x}$