Exponential Growth and Decay: Formula, Graphs and Examples

Q: How do I turn a percentage growth rate into a growth factor?

Add the rate as a decimal to 1. Growth of 8\% gives 1.08; decay of 8\% gives 0.92.

How multiplying by a constant factor differs from adding a constant amount, how to read $a \cdot b^x$ , how percentage growth turns into a growth factor, and why exponential growth overtakes everything in the end.

4 min read Supports 4 lessons

Suppose you are offered a job paying £100 more each day, or one paying 1p on day one and doubling every day after. The first looks generous. By day 30, the second is paying over £5 million a day.

The first is linear: it adds the same amount each step. The second is exponential: it multiplies by the same factor each step.

a_{1}

a_{2}

a_{3}

a_{4}

a_{5}

a_{6}

a_{7}

128

a_{8}

Doubling: each bar is twice the last. Small at first, then explosive.

Exponential function

f(x) = a \cdot b^{x}

$a$ is the starting value (when $x = 0$ ) and $b$ is the growth factor.

Slide the growth factor below. Above $1$ the curve grows; below $1$ it decays; at exactly $1$ nothing changes. The dashed line is linear growth for comparison.

From percentages to growth factors

Growth of $r\%$ per step means multiplying by $1 + \frac{r}{100}$ each step; decay of $r\%$ means multiplying by $1 - \frac{r}{100}$ . These are the same multipliers used for percentage change, applied again and again.

$500 \text{ people, growing } 8\% \text{ a year}$
Find the population after 3 years.

1/4

Tip

Growth of $r\%$ → factor $1 + \frac{r}{100}$ . Decay of $r\%$ → factor $1 - \frac{r}{100}$ . A car losing $15\%$ of its value each year has factor $0.85$ .

Decay and half-life

Radioactive material, the temperature gap of a cooling cup of tea and the value of a car all decay exponentially. A handy way to describe decay is the half-life: the time it takes for the amount to halve. Each half-life multiplies what is left by $\frac12$ .

Every 2 units of time, half of what is left disappears.

Exponential beats linear

Key idea

However steep a straight line is, an exponential with $b > 1$ eventually overtakes it. Doubling $30$ times turns $1$ into more than a billion.

Compound interest

A savings account paying $5\%$ a year, compounded annually, grows by the factor $1.05$ each year. After $t$ years, £ $P$ becomes $P \cdot 1.05^t$ . Interest earns interest, which is what makes the growth exponential rather than linear.

Worked example

£2000 is invested at $3\%$ a year. How much is there after 10 years?

1
Growth factor $1.03$ .
$2000 \times 1.03^{10}$

To find when something reaches a target, such as how long until the money doubles, you need to solve for an exponent. That is exactly what logarithms are for.

Watch out

$2^x$ and $x^2$ are different. In $x^2$ the variable is the base; in $2^x$ it is the exponent. Exponentials grow far faster.

Common questions

What is the difference between linear and exponential growth?: Linear growth adds the same amount each step, so it makes a straight line. Exponential growth multiplies by the same factor each step, so it starts slowly and then rises ever faster.
How do I turn a percentage growth rate into a growth factor?: Add the rate as a decimal to $1$ . Growth of $8\%$ gives $1.08$ ; decay of $8\%$ gives $0.92$ .

Practise exponential growth and decay

The Exponential Growth & Decay lesson has worked examples and unlimited practice that adapts to you.

Open the lesson

Common questions

What is the difference between linear and exponential growth?

Linear growth adds the same amount each step, so it makes a straight line. Exponential growth multiplies by the same factor each step, so it starts slowly and then rises ever faster.

How do I turn a percentage growth rate into a growth factor?

Add the rate as a decimal to $1$ . Growth of $8\%$ gives $1.08$ ; decay of $8\%$ gives $0.92$ .