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MathematicsGCSE Higher & A Level

What is exponential growth and decay?

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

Exponential growth and decay explained

The model y=Abty = Ab^{t} has b>1b > 1 for growth and 0<b<10 < b < 1 for decay. Compound interest, population growth, radioactive decay and cooling all fit this shape.
At A Level the model is usually written with base ee, and logarithms are used to solve for the time when a target value is reached.

Key formula

y=Aekty = A e^{kt}

Worked example

£2000 at 4% compound interest for 5 years grows to 2000×1.045≈£24332000 \times 1.04^{5} \approx £2433.

Examiner tip

Read whether the rate is per year or per month and match the exponent to it — mismatched units are the top source of lost marks here.

Exponential growth and decay: common questions

What is a half-life?
The time taken for a decaying quantity to halve. It is constant for exponential decay regardless of the starting amount.
How do I find the rate constant k?
Substitute a known data point into the model and take natural logs to solve for k.

Related terms

Logarithm

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

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.

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