Definition
Definition
The logarithm to base $a$ of $b$ is the number $c$ such that $a^{c} = b$:
${\color{#e11d48}a}$ is the base, ${\color{#059669}b}$ is the argument and ${\color{#d97706}c}$ is the result.
For it to make sense, the base and the argument must satisfy:
The idea
Taking a logarithm means asking what power you have to raise the base to in order to get the argument.
Why these conditions: a negative base can't be raised to any exponent ($(-2)^{1/2}$ is not real), and base $1$ is useless because $1^{c} = 1$ always. And since a power with a positive base is always positive, the argument must be positive: the logarithm of $0$ or of a negative number does not exist.
Examples
Examples
| Logarithm | Why |
|---|---|
| $\log_{2} 8 = 3$ | $2^{3} = 8$ |
| $\log_{10} 100 = 2$ | $10^{2} = 100$ |
| $\log_{4} 64 = 3$ | $4^{3} = 64$ |
| $\log_{3} \frac{1}{9} = -2$ | $3^{-2} = \frac{1}{3^{2}} = \frac{1}{9}$ |
| $\log_{\frac{2}{3}} \frac{3}{2} = -1$ | $\left(\frac{2}{3}\right)^{-1} = \frac{3}{2}$ |
| $\log_{5} 5 = 1$ | $5^{1} = 5$ |
| $\log_{5} 1 = 0$ | $5^{0} = 1$ |
| $\log_{5} 0$ | does not exist: no power of $5$ equals $0$ |
Worked example
When the answer isn't obvious, rewrite it as a power and put everything in the same base:
If the bases are equal, the exponents must be equal too.
Types of logarithm
| Type | Notation | Base |
|---|---|---|
| General | $\log_{a} b$ | $a$ |
| Common (decimal) | $\log b$ | $10$ |
| Natural (Napierian) | $\ln b$ | $e \approx 2.718$ |
When no base is written, it is $10$. The last two are the ones on the calculator, under the log and ln keys.
Examples: the unknown in the base or in the argument
If the unknown is the base:
If the unknown is the argument: