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Exponential Growth or Decay Calculator

Growth or decay proportional to size.

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What does Exponential Growth mean?

Exponential growth describes a process where a quantity changes by a fixed percentage over equal time periods. Unlike linear growth, where a constant amount is added each period, exponential growth compounds — the larger the value becomes, the faster it grows. This pattern appears everywhere: population growth, compound interest, viral spread, and technology adoption.

Exponential decay is the same process running in reverse: the quantity loses a fixed percentage of whatever is left each period, so it falls quickly at first and then ever more slowly, approaching zero without ever quite reaching it. Radioactive half-lives, asset depreciation, drug clearance, and a cooling cup of coffee all follow this curve. Use the Growth / Decay toggle to pick the direction — everything else about the calculation is identical.

How to calculate Exponential Growth

Exponential growth uses the formula Final Value = Initial Value × (1 + r)^n, where r is the growth rate per period as a decimal and n is the number of periods. Starting with 1,000 at a 5% growth rate over 10 periods: 1,000 × (1.05)^10 = 1,628.89. The growth factor is Final Value / Initial Value — about 1.63x here.

Exponential decay applies the same rate negatively: Final Value = Initial Value × (1 − r)^n. That same 1,000 falling 5% per period for 10 periods becomes 1,000 × (0.95)^10 = 598.74, a decrease of 401.26 with 59.87% of the original value remaining. In decay mode the rate has to stay under 100%, because losing everything in a single period leaves nothing left to decay.

Worked example: population growth

Say a town of 50,000 people grows 2% a year. Enter 50,000 as the initial value, 2 as the growth rate and 10 as the number of periods (years): 50,000 × (1.02)^10 = 50,000 × 1.2190 = 60,950 people. That is an increase of 10,950 residents over the decade, a growth factor of 1.22x — noticeably more than the 10,000 you would get by naively adding 2% of the starting population (1,000 people) ten times, because each year's growth is calculated on the larger population left by the year before.

Switch the same numbers to Decay and you model a shrinking town: 50,000 × (0.98)^10 = 40,854 people, a loss of 9,146 residents with 81.71% of the original population remaining.

FAQ

Both change a value by a fixed percentage each period; only the direction differs. Growth multiplies by (1 + r) per period, so the value accelerates upward. Decay multiplies by (1 − r), so the value falls by a shrinking absolute amount each period and approaches zero. Use the Growth / Decay toggle to switch between them — the growth factor tells you the direction too: above 1.0x is growth, below 1.0x is decay.

To find the average annual growth rate from two population counts, use r = (final ÷ initial)^(1/n) − 1, where n is the number of years, then multiply by 100 for a percentage. A town that went from 50,000 to 60,950 people in 10 years grew at (60,950 ÷ 50,000)^(1/10) − 1 = 1.2190^0.1 − 1 = 0.02, or 2% a year. Put that 2% back into this calculator with 50,000 as the initial value to project the population forward.

The natural increase rate is births minus deaths in a population over a year, conventionally expressed per 1,000 people: 18 births and 6 deaths per 1,000 residents gives a natural increase rate of 12 per 1,000, or 1.2%. It deliberately ignores migration, so it is not the same thing as the overall population growth rate. This calculator does not compute it for you — it compounds a rate you already know — but once you convert a natural increase rate to a percentage (divide by 10) you can enter it as the growth rate to project a population that neither gains nor loses migrants.

Linear growth adds a constant amount each period (e.g., +100 per year), producing a straight line. Exponential growth multiplies by a constant factor each period (e.g., ×1.05 per year), producing a curve that starts slowly and accelerates. Over long time horizons, exponential growth vastly outpaces linear growth.

Common examples include compound interest on investments, population growth, bacterial reproduction, viral content spread, and Moore's Law in computing. Any process where the rate of increase is proportional to the current size exhibits exponential growth.

Yes. A negative growth rate models exponential decay — the quantity shrinks by a fixed percentage each period. You can either enter a negative rate in Growth mode or enter the same figure as a positive number in Decay mode; both give the same answer. Examples include radioactive decay, depreciation of assets, and cooling of objects.

A decay rate of 100% removes the entire value in one period, leaving nothing behind, and a rate above 100% would remove more than exists — mathematically the formula would start flipping the value between positive and negative. Anything that seems to lose more than 100% per period is better modelled over shorter periods: 50% per half-year, for instance, rather than 100% per year.

Compound interest is a specific application of exponential growth. When interest is reinvested, your balance grows exponentially because each period's interest is calculated on the new, larger balance. The exponential growth formula is the foundation of all compound interest calculations.

The growth factor (Final Value / Initial Value) tells you how many times the original value the quantity has become. A growth factor of 2.0x means the value doubled, 3.0x means it tripled, and so on. In decay mode the same figure is below 1.0x — 0.82x means 82% of the starting value is left — and the calculator also shows it as a percentage remaining.

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