Logarithm Definition

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

The exponent that 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 $2$.
$$25=5^2 \Leftrightarrow 2=\log_{5}{25}$$
If $b=a^x,a \ne 1, a>0$, we say that $x$ is the logarithm in base $a$ of $b$, and then:
$$b=a^x \Leftrightarrow x = \log_{a}{b}$$
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.

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

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

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 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.

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

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