# Log Laws

## Table of Contents

Logarithms are the inverse of exponentials. They are useful for solving equations with exponents, and they are used in many other areas of mathematics. Here are some commonly used logarithm properties.

For any positive real number \( \small b \neq 1 \), and positive real numbers \( \small a \), and \( \small c \), the following properties hold:

$$ \log_b(1) = 0 $$ $$ \log_b(b) = 1 $$ $$ \log_b(a \cdot c) = \log_b(a) + \log_b(c) $$ $$ \log_b \left(\frac{a}{c} \right) = \log_b(a) - \log_b(c) $$ $$ \log_b(a^c) = c \cdot \log_b(a) $$ $$ b^{\log_b(k)} = k $$

Note that \( k \) can be any real number in the last equation.

## Change of Base Formula #

The change of base formula is useful for converting logarithms to different bases.

$$ \log_b(a) = \frac{\log_c(a)}{\log_c(b)} $$