Exponential Growth / Decay Calculator
Model exponential growth and decay over time. Project future values using initial amount, growth rate, and time period.
Reviewed by the ToolNestr Editorial Team — July 2026
How exponential growth and decay work
Exponential growth occurs when a quantity increases by a fixed percentage over each time period. The formula is y = a(1 + r)ᵗ, where a is the initial value, r is the growth rate (as a decimal), and t is the number of time periods. For decay, the formula is y = a(1 - r)ᵗ, where the quantity decreases by a fixed percentage each period.
The key characteristic of exponential growth is that the rate of increase accelerates over time — the larger the quantity becomes, the faster it grows. This snowball effect is why exponential growth is often described as "compound growth." In contrast, exponential decay approaches zero but never quite reaches it — there are always traces remaining, which is why radioactive decay half-lives are measurable but complete decay is technically infinite.
Continuous compounding uses the natural exponential function y = ae^(rt). This model assumes that growth happens at every instant rather than at discrete intervals (like annually or monthly). Continuous growth is the natural model for many physical and biological processes, including population growth, radioactive decay, and bacterial reproduction.
The formula explained
Discrete model
y = a(1 ± r)t
Use +r for growth, -r for decay. This models annual or periodic compounding like interest payments or population censuses.
Continuous model
y = aert
For growth; use y = ae^(-rt) for decay. This models natural processes like radioactive decay, bacterial growth, and continuously compounded interest.
Worked example: Investment growth
Initial investment = $1,000, Annual return = 5%, Time = 10 years
Discrete (annual compounding): y = 1000(1 + 0.05)¹⁰ = 1000 × 1.05¹⁰
1.05¹⁰ = 1.62889..., so y = 1000 × 1.62889 = $1,628.89
Continuous compounding: y = 1000 × e^(0.05 × 10) = 1000 × e^0.5
e^0.5 = 1.64872..., so y = 1000 × 1.64872 = $1,648.72
The continuous model yields about $20 more over 10 years because growth is compounded at every instant rather than once per year.
Investors
Compound interest follows exponential growth. The Rule of 72 says divide 72 by the interest rate to estimate doubling time. At 5%, money doubles in approximately 72/5 = 14.4 years (close to the exact value of 14.2 years).
Scientists
Radioactive decay follows exponential decay with a characteristic half-life. Carbon-14 dating, medical isotope decay, and nuclear waste management all use exponential decay models. The decay constant λ = ln(2) / half-life.
Biologists
Bacterial colonies, viral spread, and cell division follow exponential growth under ideal conditions. Population dynamics, drug clearance from the bloodstream, and enzyme kinetics all use exponential models extensively.
Epidemiologists
Infectious disease spread follows exponential growth in the early stages: the number of infected people doubles at regular intervals. The basic reproduction number R₀ determines whether an outbreak grows (R₀ > 1) or dies out (R₀ < 1).
How to use the exponential calculator
Enter parameters
Type the initial value, growth/decay rate (as a percentage), and the time period. Select "Growth (+)" or "Decay (−)" mode.
Compare models
Both discrete and continuous results are shown. The discrete model assumes periodic compounding; the continuous model assumes instantaneous compounding.
Check doubling/half-life
The calculator shows the doubling time (for growth) or half-life (for decay), giving you insight into the speed of change.
Tips for exponential modeling
The Rule of 72
For quick mental estimates of doubling time, divide 72 by the growth rate percentage. At 8% growth: 72/8 = 9 years (actual is 9.01). At 12%: 72/12 = 6 years (actual is 6.12). The rule is most accurate for rates between 6% and 10%.
Units must be consistent
Ensure the time unit matches the rate period. If the rate is annual (5% per year), time must be in years. A monthly rate requires time in months. Mixing units (annual rate with monthly time) produces incorrect results. The calculator assumes rate and time share the same unit.
Exponential growth is not sustainable forever
In the real world, exponential growth eventually hits constraints: limited resources, space, or market saturation. Logistic growth models account for carrying capacity. Use exponential models for short-to-medium term projections, but be aware of limitations for long-range forecasting.
The difference between discrete and continuous
As the compounding frequency increases (annual → monthly → daily → continuous), the final value approaches the continuous limit. The continuous model gives the maximum possible growth for a given rate. For most practical purposes, daily compounding is very close to continuous.
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Frequently asked questions
What is exponential growth?
Exponential growth occurs when a quantity increases at a rate proportional to its current value. The formula is y = a(1 + r)ᵗ, where a is initial amount, r is growth rate, and t is time.
What is exponential decay?
Exponential decay is the opposite of growth — a quantity decreases at a rate proportional to its current value. The formula is y = a(1 - r)ᵗ, with r positive but subtracted.
What is continuous compounding?
Continuous compounding uses the formula y = ae^(rt), where e is Euler's number (≈ 2.71828). This represents growth that compounds at every instant rather than at discrete intervals, common in natural processes.
What is the doubling time?
Doubling time is the time it takes for an exponentially growing quantity to double. It is calculated as t = ln(2) / ln(1 + r) for discrete growth, or t = ln(2) / r for continuous growth.
What is the half-life?
Half-life is the time it takes for an exponentially decaying quantity to halve. It is calculated as t = ln(0.5) / ln(1 - r) for discrete decay, or t = ln(2) / r for continuous decay (using decay constant).