📈 Exponential Growth & Decay: Calculator

Model exponential growth or decay, and find the exact doubling time or half-life.

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periods
FINAL VALUE
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after the chosen number of periods
Total Change
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Doubling Time
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Final Value = Initial Value × (1 + r)^t

How the Exponential Growth & Decay Calculator Works

Exponential growth and decay describe quantities that change by a constant percentage rate every period, rather than by a constant amount — population growth, radioactive decay, compound interest, and the spread of an epidemic can all follow this same mathematical pattern. This calculator projects a starting value forward (or backward, in relative terms) by a fixed percentage rate over any number of periods, and also computes the doubling time (for growth) or half-life (for decay) using the exact logarithmic formula, not an approximation.

Growth: Final Value = Initial Value × (1 + r)t    Doubling Time = ln(2) / ln(1 + r)
Decay: Final Value = Initial Value × (1 − r)t    Half-Life = ln(2) / (−ln(1 − r))
(r = rate as a decimal, t = number of time periods)

Doubling Time and Half-Life: What They Actually Mean

Doubling time is the number of periods it takes a growing quantity to exactly double in size at a constant growth rate — a population growing 5% per year, for instance, doubles roughly every 14.2 years. Half-life is the mirror concept for decay: the number of periods it takes a shrinking quantity to fall to exactly half its starting value. Both are computed here using the precise natural-logarithm formula rather than shortcut approximations like the "Rule of 70," so the results stay accurate even at larger growth or decay rates.

Common Real-World Applications

This model applies anywhere a quantity changes by a fixed percentage each period: population growth, bacterial or viral spread, radioactive isotope decay, depreciation of an asset's value, or the decline of a drug's concentration in the bloodstream. Note that this is a periodic (discrete) compounding model — each period's change is calculated on the value at the end of the previous period — which is the standard approach for most population, financial, and half-life calculations. For continuous, moment-by-moment compounding (used in some physics and finance contexts), a related but distinct formula using e (Euler's number) would apply instead.

📎 Related Tools: Half-Life Calculator Compound Interest Calculator Exponent Calculator Percentage Calculator

❓ Frequently Asked Questions

What is the difference between exponential growth and exponential decay?
Exponential growth means a quantity increases by a fixed percentage each period, following Final = Initial × (1 + r)^t. Exponential decay means it decreases by a fixed percentage each period, following Final = Initial × (1 − r)^t. Both use the same basic structure, just with growth or shrinkage.
What is doubling time and how is it calculated?
Doubling time is how many periods it takes a growing quantity to reach exactly twice its starting value at a constant rate. It's calculated exactly as t = ln(2) / ln(1 + r), where r is the growth rate as a decimal (e.g., 0.05 for 5%).
What is half-life and how is it calculated?
Half-life is how many periods it takes a decaying quantity to fall to exactly half its starting value at a constant decay rate. It's calculated exactly as t = ln(2) / (−ln(1 − r)), where r is the decay rate as a decimal.
Can I use this calculator for radioactive decay problems?
Yes — radioactive decay follows the same exponential decay model used here. If you know a substance's decay rate per period, this calculator will compute its remaining quantity after any number of periods, as well as its half-life.
Why does this use a percentage-per-period rate instead of continuous compounding?
This calculator models discrete, periodic compounding (the value changes by a fixed percentage at the end of each period), which is the standard approach for population models, half-life problems stated as a rate, and most finance and biology applications. Continuous compounding, which uses the mathematical constant e, is a related but different model typically used in specific physics and finance contexts.

🏆 About This Tool — Accuracy & Trust

🔒 Data Privacy: All calculations run entirely within your browser using JavaScript. Nothing you enter here is ever transmitted to our servers, stored, sold, or shared.

📐 Accuracy Note: This tool uses standard, widely published formulas and guidelines. Results are estimates for informational purposes; for financial, medical, or engineering decisions, consult a licensed professional.

📅 Last Updated: September 2026.