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.

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

Algebra Algebraic Fractions Arc Binomial Expansion Capacity Common Difference Common Ratio Differentiation Divisibility Proof Double-Angle Formula Equation Exponent Exponential Function Factorials Factorise Functions Geometric Sequence Geometric Series Index Laws Inequality Integration Kinematics Length Conversion Logarithm Logarithmic Functions Mass Conversion Mathematical Induction Measurement Perfect Square Perimeter Prime Factorisation Probability Proof Pythagoras Theorem Quadratic Quadratic Factorise Rational Functions Sequence Sketching Graphs Surds Time Transformation Trigonometric Functions Trigonometric Properties Volume


Your email address will not be published. Required fields are marked *