Logarithms explained: what they are and how to use them

TutorAura guide · Updated 9 November 2026

A logarithm answers the question: what power do I need? $\log_2 8 = 3$ because $2^3 = 8$ . In general:

\log_a x = y \quad \Longleftrightarrow \quad a^y = x

The small number $a$ is the base. Logs undo powers, in the same way that square roots undo squaring, which is why they're used to solve equations where the unknown is in the power. They're taught in A level Maths, the IB, and US Algebra 2 and Precalculus.

Key takeaways

log_a x is the power you raise a to, to get x.
log(ab) = log a + log b; log(a/b) = log a − log b; log(aⁿ) = n log a.
ln means log base e (about 2.718).
Use logs to solve equations with the unknown in the power.

Examples to build intuition

Log	Because	Value
$\log_{10} 1000$	$10^3 = 1000$	3
$\log_2 32$	$2^5 = 32$	5
$\log_5 1$	$5^0 = 1$	0
$\log_3 \tfrac{1}{9}$	$3^{-2} = \tfrac{1}{9}$	$-2$

$\log_a a = 1$ and $\log_a 1 = 0$ for any base.

The laws of logarithms

\log a + \log b = \log(ab) \qquad \log a - \log b = \log\left(\frac{a}{b}\right) \qquad \log(a^n) = n\log a

These follow from the laws of indices: multiplying powers means adding the indices, so multiplying numbers means adding their logs. (All the logs must have the same base.) Revise indices with our laws of indices questions.

Solving exponential equations

Example: $2^x = 10$ .

Take logs of both sides: $\log 2^x = \log 10$ , so $x \log 2 = \log 10$ , and $x = \dfrac{\log 10}{\log 2} \approx 3.32$ .

Check: $2^{3.32} \approx 10$ . The same method finds how long an investment takes to double, or how long a radioactive sample takes to decay.

Natural logs and e

$e \approx 2.718$ is a special base that appears in growth and decay. The log to base $e$ is the natural log, written $\ln$ : $\ln e = 1$ and $\ln 1 = 0$ . Equations like $e^{2x} = 7$ are solved with ln: $2x = \ln 7$ , so $x \approx 0.973$ .

Changing the base

Calculators usually have log (base 10) and ln (base $e$ ) buttons. For any other base:

\log_a x = \frac{\log x}{\log a}

So $\log_2 10 = \log 10 \div \log 2 \approx 3.32$ , matching the example above.

Common mistakes

Thinking $\log(a + b) = \log a + \log b$ (only products split).
Dividing logs instead of subtracting: $\tfrac{\log a}{\log b}$ is not $\log\tfrac{a}{b}$ .
Taking the log of a negative number or zero: it's undefined.

Logs connect to compound growth: see our compound interest calculator and exponential growth and decay questions.

Where logarithms are used

Logarithmic scales compress huge ranges into manageable numbers:

pH: $\text{pH} = -\log_{10}[\text{H}^+]$ , so each pH unit is a tenfold change in hydrogen ion concentration.
Sound: decibels are logarithmic: 10 dB more means ten times the sound intensity.
Earthquakes: each step up on a magnitude scale is about ten times the ground movement.

Logs also solve growth and decay problems, such as how long a population takes to double.

Practice questions

Find $\log_3 81$ .
Simplify $\log 4 + \log 25$ (base 10).
Solve $3^x = 20$ , to 3 significant figures.
Find $\log_2 \tfrac{1}{8}$ .

Answers: 1. 4 (because $3^4 = 81$ ). 2. $\log 100 = 2$ . 3. $x = \log 20 \div \log 3 \approx 2.73$ . 4. $-3$ .

Logs and indices side by side

Index form	Log form
$2^5 = 32$	$\log_2 32 = 5$
$10^{-2} = 0.01$	$\log_{10} 0.01 = -2$
$e^0 = 1$	$\ln 1 = 0$

Frequently asked questions

What is a logarithm?

The power you need to raise the base to, to get a number: log₂ 8 = 3 because 2³ = 8.

What are the laws of logarithms?

log a + log b = log(ab), log a − log b = log(a/b), and log(aⁿ) = n log a, all with the same base.

What is ln?

The natural logarithm: the log to base e, where e is about 2.718.

Sources

Cambridge International AS & A Level Mathematics 9709. Cambridge International.

Laws of indices questions and answers

8 GCSE and IGCSE questions with worked answers, checked as you type.

Exponential growth and decay questions and answers