Exponential Growth and Decay: Formula, Doubling Time and Half-Life

Exponential growth happens when something increases by the same percentage every period, and exponential decay when it decreases by the same percentage. Both follow $A = A_0(1 + r)^t$, where $r$ is the rate as a decimal (negative for decay). The time to double, or halve, comes from logarithms.

Linear vs exponential

Adding 8 each year is linear growth. Multiplying by 1.08 each year is exponential growth. Early on they look similar, but exponential growth eventually races ahead because each increase is bigger than the last.

The formula

Worked example 1: growth

500 bacteria grow by 8% per hour. After 10 hours:

Worked example 2: decay

80 g of a medicine leaves the body at 15% per hour. After 6 hours:

Doubling time and half-life

Solve $(1 + r)^t = 2$ for doubling, or $= \frac{1}{2}$ for halving:

At 8% growth, $t = \frac{\ln 2}{\ln 1.08} \approx 9.0$ periods. At 15% decay, the half-life is $\frac{\ln 0.5}{\ln 0.85} \approx 4.27$ hours.

Common mistakes

• Using 8 instead of 0.08 for the rate.
• Using $1 - r$ when $r$ is already negative.
• Thinking a 50% rise then a 50% fall gets you back to the start (it leaves 75%).

Practice questions

1. 1,000 grows by 5% a year. How much after 3 years?
2. A car worth 20,000 loses 12% a year. What is it worth after 2 years?
3. What is the doubling time at 10% growth?

Answers: 1) 1,157.63 2) 15,488 3) about 7.27 periods

The exponential growth calculator draws the curve and marks the doubling time or half-life. For money with interest added several times a year, use the compound interest calculator. Logarithms are explained in logarithms for beginners.

Frequently asked questions

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