Chapter 4: Logarithms

The Logarithmic Operation

Let $a$ and $b$ be positive real numbers. If $a>0$ and $a \neq 1$, we define the logarithm of $b$ with base $a$ by $$ \log_{a}(b)=x \Leftrightarrow a^{x} = b. $$ That is, $\log_{a}(b)$ is the solution of the equation $a^{x}=b$.

Thus, the logarithmic function $y=\log_{a}(x)$, $(x>0)$ is the inverse function of $f(x) = a^{x}$ (more in inverse functions). The logarithmic and exponential operations are therefore inverse operations and satisfy the following properties of inverse functions.

1. $\log_{a}(a^{x})=x \quad (x \in \mathbb{R}$)
2. $a^{\log_{a}(x)}=x \quad (x > 0)$