Exponential Decay Calculator
Calculate exponential decay. Enter starting value, decay rate per period and time to see remaining amount, amount decayed, half-life and a decay chart.
How to use this tool
- Enter initial value, decay rate per period and number of periods in the fields above.
- Results update instantly as you type — or click Calculate.
- Read your remaining value and the full breakdown beneath it.
Exponential decay follows A = A₀(1 − r)^t, where A₀ is the initial amount, r the decay rate per period, and t the number of periods. The half-life is log(0.5) / log(1−r).
Formula
Remaining = P × (1 − d)t
Half-life = ln(0.5) ÷ ln(1 − d)
Where P = initial value, d = decay rate per period (decimal), t = number of periods.
How it works
This calculator applies the discrete period-by-period decay formula, reducing the value by a fixed percentage each period. The half-life is the number of periods for the value to reach half its starting amount, solved from (1 − d)n = 0.5. The model assumes a constant decay rate every period; for continuously varying processes (e.g., radioactive isotopes measured in continuous time), use a half-life-based model instead.
Worked example
- Initial value P = 1000, decay rate d = 10% = 0.10 per period, t = 10 periods.
- Apply the formula: Remaining = 1000 × (1 − 0.10)^10 = 1000 × 0.90^10.
- 0.90^10 = 0.348678..., so Remaining = 1000 × 0.348678 = 348.678.
- Amount decayed = 1000 − 348.678 = 651.322.
- Half-life = ln(0.5) / ln(0.90) = −0.6931 / −0.10536 ≈ 6.579 periods.
Remaining value: 348.678; Amount decayed: 651.322.
Common mistakes to avoid
- Entering the decay rate as a whole number percentage (e.g. 10 for 10%) instead of a decimal (0.10), which models 1,000% decay per period and collapses the value to near zero immediately.
- Confusing decay rate with half-life — a 10% per period decay rate does not mean the value halves every 10 periods; the actual half-life is ln(0.5)/ln(0.9) approximately 6.6 periods.
- Applying the continuous formula to a quantity that decays discretely (e.g. items removed per batch), producing a smooth curve when the real process is stepped.
Key terms
- Decay rate per period (d)
- The percentage of the current value lost in each period, expressed as a decimal (e.g., 10% → 0.10).
- Half-life
- The number of periods after which the remaining value falls to exactly half the initial value.
- Remaining value
- The quantity that persists after t periods of exponential decay.
- Amount decayed
- The total quantity lost: initial value minus remaining value.
- Discrete vs continuous decay
- Discrete decay reduces by a fixed fraction each period; continuous decay uses e^(−kt) and is common in physics for radioactive substances.
Frequently asked questions
- What is the difference between decay rate and half-life?
- The decay rate r is the fraction lost each period; the half-life is the number of periods until only half remains. They are linked by t½ = log(0.5)/log(1−r).
- Does this assume continuous or discrete decay?
- This calculator uses discrete (period-by-period) decay. For continuous decay use A = A₀·e^(−λt) with the half-life calculator.