What is the exponential decay formula?

A = A₀ × e^(−rt), where A₀ is the initial amount, r is the continuous decay rate, t is elapsed time, and e is Euler's number (≈2.71828).

Half-life is the time it takes a decaying quantity to reduce to exactly half of whatever amount it currently is, calculated as ln(2) / r. It's constant regardless of the starting amount — a substance always takes the same time to halve, no matter when you start measuring.

How is the decay rate different from the half-life?

The decay rate (r) is the continuous rate of decrease used directly in the exponential formula. The half-life is a derived, more intuitive quantity — the two are related by half-life = ln(2)/r, so either one determines the other.

Can the remaining amount ever reach exactly zero?

No — exponential decay approaches zero asymptotically but mathematically never reaches it in finite time. In practice, the remaining amount becomes negligibly small after several half-lives.

How is this related to the Half Life Calculator?

This calculator projects forward from a known continuous rate to find the remaining amount and half-life. The Half Life Calculator instead solves for any one of the initial amount, remaining amount, time, or half-life given the other three — useful when you know the half-life but not the underlying rate.

What's the difference between decay rate and decay percentage?

The continuous decay rate (r) used in e^(−rt) is not the same as a simple percentage decrease per period — a 10% decrease per year corresponds to a continuous rate of −ln(0.9) ≈ 0.105, not exactly 0.10, because continuous decay compounds constantly.

Why is e used instead of a simple percentage multiplier?

Using e^(−rt) models continuous, instant-by-instant decay rather than decay that only happens at discrete steps, which is the standard approach in physics and calculus-based models like radioactive decay.

Where is exponential decay used in real life?

It models radioactive isotope decay, drug concentration decreasing in the bloodstream over time, cooling objects approaching room temperature, and the declining value of a depreciating asset.

Exponential Decay Calculator — Find Remaining Amount and Half-Life

Our Exponential Decay Calculator computes continuous exponential decay using A = A₀e^(−rt), where A₀ is the initial amount, r is the continuous decay rate, and t is elapsed time. It also computes the half-life (ln(2)/r) — how long it takes any decaying quantity to reduce to half its current value, regardless of the starting amount. Useful for radioactive decay, drug elimination from the body, depreciation, and any quantity that shrinks proportionally to its current size.

Quick Answer

Exponential decay follows A = A₀e^(-rt), where A₀ is the starting amount, r is the decay rate, and t is time. Enter your values below to get the remaining amount and the half-life (ln(2)/r) for that decay rate.

Initial amount (A₀)

Decay rate (r)

Time (t)

Enter an initial amount, rate, and time, then click Calculate.

How to Use the Exponential Decay Calculator — A = A₀e^(-rt) Online

1
Enter the initial amount (A₀).
2
Enter the continuous decay rate (r) as a decimal, e.g. 0.05 for 5%.
3
Enter the elapsed time (t).
4
Click 'Calculate' to get the remaining amount and the half-life for this decay rate.

Why Use Exponential Decay Calculator — A = A₀e^(-rt) Online?

Exponential decay describes any quantity that shrinks at a rate proportional to how much remains — the same mathematical shape underlying radioactive isotopes, the elimination of medication from the bloodstream, and the fading value of a depreciating asset. The formula A = A₀e^(−rt) mirrors exponential growth with a negated rate, and the half-life formula ln(2)/r gives a rate-independent way to express how quickly the decay happens. Computing this by hand requires the same care with negative exponents that trips people up in exponential growth; this calculator handles both the projection and the half-life instantly.

Frequently Asked Questions

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