Exponential Growth & Decay Calculator (Doubling Time & Half-Life)
Calculate future population, compound growth, or radioactive decay using both discrete x(t) = x₀(1+r)^t and continuous x(t) = x₀e^(kt) models.
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How to Use the Exponential Growth & Decay Calculator (Doubling Time & Half-Life)
Exponential growth occurs whenever the rate of change of a quantity is directly proportional to its current size—meaning the larger a population, investment, or viral user base grows, the faster it adds new units. Mathematically, discrete periodic compounding is modeled by x(t) = x₀(1 + r)^t, while natural biological and physical systems that compound continuously follow x(t) = x₀e^(rt).
Formula and Calculation Methodology
- x₀: Initial starting quantity at time t = 0
- r: Decimal growth rate (r > 0) or decay rate (r < 0) per time period
- t: Number of elapsed time intervals (hours, days, or years)
When a negative rate is entered (r < 0), the same equation models exponential decay—used across pharmacology (drug elimination half-life), physics (carbon-14 radioactive decay), and asset depreciation. Our calculator outputs both discrete and continuous trajectories alongside the exact logarithmic Doubling Time (ln(2) ÷ ln(1 + r)) or Half-Life.
Doubling Time & 10-Period Growth Factor by Rate (%)
| Growth Rate (r) | Exact Doubling Time | Rule of 70 Estimate | 10-Period Multiplier | 20-Period Multiplier |
|---|---|---|---|---|
| 2% | 35.00 periods | 35.0 periods | 1.219x | 1.486x |
| 5% | 14.21 periods | 14.0 periods | 1.629x | 2.653x |
| 7% | 10.24 periods | 10.0 periods | 1.967x | 3.870x |
| 10% | 7.27 periods | 7.0 periods | 2.594x | 6.727x |
| 15% | 4.96 periods | 4.7 periods | 4.046x | 16.367x |
Frequently Asked Questions
When should I use the discrete formula vs. the continuous e^(rt) formula?
Use discrete growth x₀(1 + r)^t when growth is tallied at fixed intervals (such as annual interest or yearly census counts). Use continuous growth x₀e^(rt) for bacterial division, radioactive decay, or continuous compounding.
How does the Rule of 70 estimate doubling time?
Because ln(2) ≈ 0.693, dividing 70 by the percentage growth rate (70 ÷ R%) gives a fast mental approximation of how many periods it takes a quantity to double.
Can exponential decay ever reach exactly zero?
Mathematically, an exponential decay curve approaches zero asymptotically (getting infinitely close without touching 0), though in physical systems discrete atoms or molecules eventually reach zero.
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