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. Also try the Logarithm Calculator (log).
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. See also our log_a(x) Result — Change of Base.
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. You might also find our calculate Log Base 2 (log₂x), Log Base 10 (log₁₀x) & Natural Log (ln x) — Log Base 2 useful.
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.