Enter a starting value, a decay rate, and a number of periods to see the remaining value, percent remaining, and half-life update as you type.
| Period t | Value x(t) | Change |
|---|---|---|
| 0 | 500.00 | N/A |
| 1 | 475.00 | -25.00 |
| 2 | 451.25 | -23.75 |
| 3 | 428.69 | -22.56 |
| 4 | 407.25 | -21.43 |
| 5 | 386.89 | -20.36 |
| 6 | 367.55 | -19.34 |
| 7 | 349.17 | -18.38 |
| 8 | 331.71 | -17.46 |
| 9 | 315.12 | -16.59 |
| 10 | 299.37 | -15.76 |
What the table shows: at the default settings, 500 units losing 5% per period for 10 periods, the value falls quickly at first, dropping by 25.00 units in period 1, then more slowly as the base shrinks, dropping by only 15.42 units in period 10. The curve never turns negative and never quite reaches zero, finishing at 299.37 units after 10 periods. 59.9% of the original amount remains.
Exponential decay is a decrease in which a quantity loses a fixed percentage of its current value every period, so the absolute amount lost shrinks over time.
A 640-gram sample losing 3% per hour loses less material in its tenth hour than its first, even though the percentage never changes, because 3% of a smaller mass is a smaller mass. This is the defining property of exponential decay, and it is the exact mirror image of exponential growth.
Plotted against time, exponential decay traces a falling curve that drops steeply at first and then flattens as it approaches the horizontal axis. The curve gets closer and closer to zero without ever touching it, a property called asymptotic decay. Anyone who has used the exponential growth calculator on this site has already seen this same curve: entering a negative rate there produces exactly the falling shape shown above.
The engine underneath is the decay factor: the fixed number the quantity is multiplied by every period. A 5% decay rate means a decay factor of 0.95, applied once per period. Ten periods of that produce a total multiplier of 0.9510 = 0.5987, meaning just under 60% of the original quantity survives, far more than the 50% a rough mental estimate of "5% times 10 periods" would suggest, because each period's loss is computed on a shrinking base.
Exponential decay applies whenever a process removes a constant proportion, not a constant amount, of whatever remains. A tank draining a fixed number of litres per minute is not decaying exponentially, however large the number. A radioactive sample losing a fixed percentage of its atoms per year is.
The exponential decay formula is x(t) = x0(1 − r)t for discrete periods, and x(t) = x0e−kt for continuous decay. Both describe the same falling curve; they differ only in how often the loss is applied.
Use this when loss happens in identifiable steps: an annual depreciation charge, a yearly population census, a daily inventory count.
Use this when decay has no natural step size: radioactive decay, cooling, or any process losing material smoothly rather than in batches.
| Symbol | Name | What it represents | Example |
|---|---|---|---|
| x(t) | Final value | The quantity remaining after t periods. | 299.37 units |
| x0 | Initial value | The quantity at t = 0, before any decay. | 500 units |
| r | Periodic decay rate | The proportional loss per period, as a decimal. | 0.05 (that is 5%) |
| k | Decay constant | The instantaneous rate used in the continuous form. | 0.0513 per period |
| t | Elapsed time | How many periods have passed. | 10 periods |
What the table shows: five symbols cover everything either form of the decay formula needs. Only x0, r, and t are ever entered by hand. x(t) is the computed result, and k is derived from r through k = −ln(1 − r) rather than supplied directly. The r-to-k conversion tool performs that conversion on its own.
To calculate exponential decay, convert the percentage rate to a decimal, subtract it from 1 to get the decay factor, raise that factor to the power of the elapsed periods, then multiply by the initial value.
Identify x₀, the quantity present before any decay has happened.
Divide the percentage rate by 100. A rate of 4% per period becomes r = 0.04.
Calculate 1 − r. With r = 0.04 the decay factor is 0.96.
Compute (1 − r)^t. With a factor of 0.96 across 12 periods, 0.96^12 = 0.612710.
Multiply x₀ by the result of step 4 to get x(t).
A warehouse holds 850 units of a perishable item and loses 4% of the remaining stock to spoilage every week. The decay formula gives the stock level after 12 weeks.
The warehouse loses 329.20 units to spoilage over the 12 weeks, leaving 61.3% of the original stock, noticeably more than the 48% a flat "4% times 12 weeks" estimate would predict.
Exponential decay and exponential growth are the same equation with opposite rate signs: growth uses a positive r and a factor above 1, while decay uses a negative r and a factor between 0 and 1. Nothing else about the formula changes.
This is why a single tool can handle both directions. Rather than building two separate calculators, the homepage's exponential growth calculator accepts a signed rate directly. Type a positive number for growth, or a negative one to calculate decay, and the same formula, the same live table, and the same plotted curve adjust automatically.
The two directions do behave differently at their limits. Growth has no ceiling: as t increases, x(t) increases without bound. Decay has a floor it approaches but never reaches. The curve gets arbitrarily close to zero but stays positive for every finite t. Any exponential rate calculator worth using should make this asymmetry visible rather than hide it, which is why the calculator on this page reports a half-life instead of a doubling time the moment the rate describes a loss.
Exponential decay is a decrease in which a quantity loses a fixed percentage of its current value every period, so the absolute amount lost shrinks as the quantity itself shrinks. A sample losing 5% per day loses less material on day 10 than on day 1, because 5% of a smaller amount is a smaller amount.
Plotted against time, exponential decay traces a falling curve that drops quickly at first and then flattens as it approaches zero, never quite reaching it.
The exponential decay formula is x(t) = x₀(1 − r)t for discrete periods, and x(t) = x₀e−kt for continuous decay. x₀ is the initial value, r is the periodic decay rate as a decimal, k is the continuous decay constant, and t is elapsed time.
Both forms are the growth formula with the sign of the rate reversed. Decay is not a separate equation, just a negative one.
Decay differs from growth only in the sign of the rate: growth multiplies by a factor above 1, while decay multiplies by a factor between 0 and 1. The same formula, x(t) = x₀(1 + r)t, produces decay automatically the moment r is negative.
Because the two share one equation, the main exponential growth calculator can compute a decay curve directly just by accepting a negative rate.
The decay constant k is the continuous-time rate in x(t) = x₀e−kt, and it relates to the periodic decay rate r through k = −ln(1 − r). A 5% periodic decay rate corresponds to a continuous decay constant of −ln(0.95) = 0.0513 per period.
The decay rate calculator converts between r and k directly, including from a known half-life.
Exponential decay never reaches exactly zero in the pure mathematical model. It approaches zero asymptotically, getting arbitrarily close without arriving. Repeatedly multiplying by a factor like 0.95 always leaves a smaller positive amount, no matter how many periods pass.
In practice a decay calculation is usually stopped once the remaining amount falls below a meaningful threshold, such as one detectable unit.
You calculate half-life from a decay rate with t½ = ln(0.5) / ln(1 − r) for discrete decay, or t½ = ln(2) / k for continuous decay. Both give the time needed for exactly half the original quantity to remain.
The dedicated half-life calculator solves this directly, including from two known data points instead of a rate.
The decay rate cannot exceed 100% in the discrete model, because a rate of exactly 100% already reduces the growth factor to zero and removes the entire quantity in one period. A rate above 100% would require the factor (1 − r) to go negative, which has no meaningful decay interpretation.
The continuous form has no such ceiling, since e−kt stays positive for any k and t.
Real-world processes that follow exponential decay include radioactive isotope decay, cooling toward room temperature, and the fading loudness of a struck bell. Each loses a fixed proportion of its current value per unit of time rather than a fixed absolute amount.
The bacterial growth calculator models the mirror-image process, the same mathematics running in the growth direction instead of decay.