# Logarithm Definition

A logarithm determines “$\textit{How many of this number do we multiply to get the number?}$”.

The exponent gives the power to which a base is raised to make a given number.
For example, $5^2=25$ indicates that the logarithm of $25$ to the base $5$ is $2$.
$$\large 25=5^2 \Leftrightarrow 2=\log_{5}{25}$$
If $b=a^x,a \ne 1, a>0$, we say that $x$ is the logarithm in base $a$ of $b$, and then:
$$\large b=a^x \Leftrightarrow x = \log_{a}{b}$$
It is read as “$b=a^x$” if and only if $x = \log_{a}{b}$.
It is a short way of writing:

If $b=a^x$ then $x=\log_{a}{b}$, and if $x=\log_{a}{b}$ then $b=a^x$.

These mean that $b=a^x$ and $x=\log_{a}{b}$ are $\textit{equivalent}$ or $\textit{interchangeable}$ statement.

\begin{align} \require{AMSsymbols} \require{color} y &= a^x \cdots (1) \\ x &= \log_{a}{y} \cdots (2) \\ \therefore \color{green}x &\color{green}= \color{green}\log_{a}{a^x} &\text{ by } (1) \text{ and } (2) \\ \end{align}

\begin{align} x &= a^y \cdots (3) \\ y &= \log_{a}{x} \cdots (4) \\ \therefore \color{green}x &\color{green}= \color{green}a^{\log_{a}{x}} &\text{ by } (3) \text{ and } (4) \end{align}

## Example 1

Write $3^2=9$ in an equivalent logarithmic statement.

$3^2=9 \leadsto 2=\log_{3}{9}$

## Example 2

Write $3=\log_{4}{64}$ in an equivalent exponential logarithmic statement.

$3=\log_{4}{64} \leadsto 4^3=64$

## Example 3

Find $\log_{3}{81}$ without using a calculator.

\begin{align} \displaystyle \log_{3}{81} &= \log_{3}{3^4} \\ &= 4 \end{align}

## Example 4

Find $\log_{5}{0.2}$ without using a calculator.

\begin{align} \displaystyle \log_{5}{0.2} &= \log_{5}{\frac{2}{10}} \\ &= \log_{5}{\frac{1}{5}} \\ &= \log_{5}{5^{-1}} \\ &= -1 \end{align}