Lesson 19.4: What is a Logarithm?

Property

For $b > 0$, $b \neq 1$, the base $b$ logarithm of of $x$, written $\log_b x$, is the exponent to which $b$ must be raised in order to yield $x$.

Examples

• $\log_3 9 = 2$ because $3^2 = 9$.
• $\log_5 125 = 3$ because $5^3 = 125$.
• $\log_4 \frac{1}{16} = -2$ because $4^{-2} = \frac{1}{16}$.

Explanation

A logarithm answers the question: "What exponent do I need to put on the base to get this number?" It's a tool for finding unknown powers. Think of it as the inverse of an exponent.

Property

For any base $b > 0$, $b \neq 1$,

$\log_b b = 1$ because $b^1 = b$

$\log_b 1 = 0$ because $b^0 = 1$