Exponential Growth and Decay Calculator
Computes a final amount after exponential growth or decay using A = Pe^(rt). Use it for population projections, radioactive half-life problems, or continuously compounded investment calculations.
About this calculator
Exponential growth and decay are modeled by the formula A = Pe^(rt), where P is the initial amount, r is the continuous growth rate (positive) or decay rate (negative), t is time, and e ≈ 2.71828 is Euler's number. For decay problems the formula becomes A = Pe^(−rt), meaning the exponent is negated. The rate r is typically expressed as a decimal (e.g., 5% = 0.05). Because the rate is in the exponent, even a small change in r produces dramatically different results over long time periods — this is the hallmark of exponential behavior. The model is continuous, unlike discrete compound interest which applies the rate at fixed intervals.
How to use
A bacterial culture starts with P = 500 cells and grows at a continuous rate of r = 3% per hour. Find the population after t = 4 hours. Set calculation_type to 'growth'. A = 500 × e^(0.03 × 4) = 500 × e^(0.12). e^(0.12) ≈ 1.1275. A ≈ 500 × 1.1275 = 563.75 cells. Enter initial_amount = 500, growth_rate = 0.03, time = 4, and select growth to confirm the result of approximately 564 cells.
Frequently asked questions
What is the difference between exponential growth and exponential decay?
Exponential growth occurs when the rate r is positive, causing the quantity to increase without bound over time — examples include unchecked population growth or money earning compound interest. Exponential decay occurs when r is negative (or the formula uses −r), causing the quantity to shrink toward zero — examples include radioactive decay or drug concentration in the bloodstream. Both processes follow the same underlying formula A = Pe^(rt); only the sign of the exponent differs, producing mirror-image curves on a graph.
How do I find the half-life of a substance using the exponential decay formula?
Half-life (t½) is the time it takes for a quantity to reduce to half its initial value. Set A = P/2 in the decay formula: P/2 = Pe^(−rt½). Dividing both sides by P gives 1/2 = e^(−rt½). Taking the natural log: ln(0.5) = −rt½, so t½ = ln(2) / r ≈ 0.693 / r. For example, if the decay rate is r = 0.02 per year, then t½ = 0.693 / 0.02 = 34.7 years. You can rearrange the same relationship to find r if you know the half-life experimentally.
Why is the continuous growth model A = Pe^(rt) used instead of simple compound interest?
Simple compound interest applies the rate at discrete intervals (annually, monthly, etc.), while the continuous model assumes the rate is applied at every infinitesimal moment. As the compounding frequency approaches infinity, the discrete formula (1 + r/n)^(nt) converges to e^(rt). The continuous model is preferred in biology, physics, and finance theory because it produces cleaner mathematics and matches processes that genuinely evolve without pause — like chemical reactions or population dynamics — more accurately than any discrete approximation.