Enter your initial value, decay rate, and elapsed time to calculate the final value after exponential decay. The Exponential Decay Calculator uses the formula x(t) = x₀ × (1 + r)ᵗ to show you the remaining quantity, total amount lost, and half-life — perfect for modeling radioactive decay, population decline, or depreciation.
Final Value x(t)
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Amount Decayed
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Percent Remaining
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Half-Life
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Exponential decay describes the process by which a quantity decreases at a rate proportional to its current value. Each time period, the same percentage is removed — but because that percentage applies to an ever-smaller amount, the absolute reduction gets smaller over time. Common examples include radioactive decay, drug elimination from the body, and asset depreciation.
This calculator uses the formula x(t) = x₀ × (1 + r)ᵗ, where x₀ is the initial value, r is the decay rate (expressed as a negative decimal, e.g. −0.05 for 5% decay), and t is the number of elapsed time periods. The result x(t) is the remaining value after t periods.
Enter your initial value, your decay rate as a positive percentage, and the number of time periods. The calculator converts the rate to negative (since it's a decay), raises (1 + r) to the power of t, and multiplies by the initial value to give you the remaining quantity.
Half-life is the amount of time it takes for a quantity to reduce to exactly half its original value. It is calculated as t₁/₂ = ln(2) / |r|, where r is the decay constant. Half-life is widely used in nuclear physics, pharmacology, and chemistry to characterize decay rates.
The decay rate (r) is the percentage lost per discrete time period and is used in the formula x(t) = x₀ × (1 − r)ᵗ. The decay constant (λ) is used in continuous exponential decay: x(t) = x₀ × e^(−λt). They are related but not identical; for small rates they are approximately equal.
Exponential decay appears in many fields: radioactive isotope half-lives in nuclear physics, drug concentration reduction in pharmacokinetics, carbon-14 dating in archaeology, population decline in ecology, cooling of objects (Newton's Law of Cooling), and depreciation of assets in finance.
If you know the initial value x₀, the final value x(t), and the time elapsed t, you can rearrange the formula to find the rate: r = (x(t)/x₀)^(1/t) − 1. The result will be negative, confirming decay. You can then express it as a percentage by multiplying by −100.
If the decay rate is 100% or more, the final value would reach zero or go negative in one step, which is not physically meaningful for most applications. This calculator caps the rate at 100% and treats any entry above that as 100% decay, resulting in a final value of 0.