Definition of a logarithm: formula, calculator and worked example

What the symbols mean

$y$: the logarithm: the exponent
$b$: the base (positive, and not 1)
$x$: the number (must be positive)

When to use it

Use it to switch between exponential and logarithmic form, which is how you solve equations where the unknown is an exponent ( $2^y = 32$ , how long until an investment doubles). Calculators only have ln and log₁₀ buttons, so for other bases combine it with the change-of-base formula.

Worked example

Find log₂ 32, the power of 2 that gives 32 (b = 2, x = 32).

Start from the formula.
$y = \log_{b} (x)$
Substitute $b$ = $2$ , $x$ = $32$ and work it out.
$y = \frac{\ln (x)}{\ln (b)}$
$= \frac{\ln (32)}{\ln (2)}$
$\approx \frac{3.46574}{0.693147}$
$= 5$
Check: $2^{5} = 32$ .

Answer log₂ 32 = 5

Common mistakes

Taking the log of zero or a negative number. $b^y$ is always positive, so $log_b x$ needs x > 0.
Reading $log_b x$ as "log times b times x". It is one function with the base as a subscript.
Using base 1. Every power of 1 is 1, so it cannot give any other number.

Calculator

See it move

What the symbols mean

When to use it

Worked example

Common mistakes