MathsGradeUp logoMathsGradeUp
MathematicsA Level & IB

What is logarithm?

A logarithm answers the question 'what power do I raise this base to?' — it is the inverse of exponentiation.

Logarithm explained

The laws of logs turn products into sums, quotients into differences and powers into multipliers, which is what makes them useful for solving exponential equations.
The natural logarithm, ln⁡\ln, has base ee and pairs with exponential growth and decay models throughout A Level and IB.

Key formula

log⁡a(xy)=log⁡ax+log⁡ay\log_{a}(xy) = \log_{a}x + \log_{a}y

Worked example

Solving 3x=203^{x} = 20: x=ln⁡20ln⁡3≈2.73x = \frac{\ln 20}{\ln 3} \approx 2.73.

Examiner tip

You cannot take the log of zero or a negative number — always check your solutions back in the original equation and reject invalid ones.

Logarithm: common questions

What is the difference between log and ln?
log usually means base 10 while ln means base e. Both obey the same laws.
Why do we plot log graphs?
Taking logs of a power or exponential relationship gives a straight line, so the gradient and intercept reveal the model's constants.

Related terms

Exponential growth and decay

Exponential growth or decay describes a quantity that changes by a fixed percentage in each equal time interval.

Differentiation

Differentiation is the process of finding the rate at which one quantity changes with respect to another — the gradient of a curve at a point.

Practise mathematics on MathsGradeUp

Take logarithm from a definition to full marks with exam-board question banks, revision notes and instant step-by-step feedback from Study Coach.

Maths revision notesPractice questionsCreate a free account

Related on MathsGradeUp

Keep studying

Practise it