What makes a change exponential
A quantity grows or decays exponentially when the change over an interval is proportional to how much is there. Not a fixed amount per period — a fixed fraction per period. That is the whole definition, and everything else follows from it.
The signature is easy to test on data. Take equally spaced observations and divide each by the one before. If those ratios are roughly constant, the process is exponential; if the differences are roughly constant instead, it is linear. A bacterial culture doubling every twenty minutes, a savings balance earning 5% a year, a radioisotope losing a fixed fraction of its remaining nuclei per second and a debt accruing interest all share that ratio-constant structure, which is why one formula serves all of them.
The consequence people underestimate is compounding over long horizons. At 5% per period, ten periods multiplies the quantity by 1.0510 = 1.6289, not by 1.5. The gap between those — 12.9 percentage points on a starting value of 1,000 — is the interest earned on interest, and it grows without bound as the horizon lengthens. Over 100 periods the same 5% multiplies by 131.5, which no amount of linear intuition prepares you for.
Decay is the same mechanism with a negative exponent. A quantity losing 50% per period retains 0.5, then 0.25, then 0.125 of the original, never reaching zero in finite time under the model even though it becomes physically negligible quickly. That asymptotic approach is why decay is described by a half-life rather than by a lifetime.
The two forms of the formula and how to convert between them
Written discretely, with the change credited once per period:
A = A₀ (1 + r)t
Written continuously, with the change happening at every instant:
A = A₀ ekt
These are the same curve, expressed with different bookkeeping. The bridge between them is the logarithm:
k = ln(1 + r) and, going the other way, r = ek − 1
At 5% per period, k = ln(1.05) = 0.0487902. The continuous rate is always smaller in magnitude than the discrete rate for growth, because continuous compounding needs a smaller instantaneous rate to reach the same year-end total. For decay the same relation holds with both quantities negative: a discrete −50% per period corresponds to k = ln(0.5) = −0.693147.
Which form you use is a matter of what your field quotes, not of what is happening physically. Finance and demography quote discrete percentages, so r is what you have. Radioactive decay, first-order chemical kinetics and RC circuit discharge are defined by a decay constant, so k is what you have. This calculator takes either and reports both, so you can move between a half-life table and a percent-per-year figure without a second tool.
Doubling time and half-life are one quantity. Set A/A₀ = 2 in the continuous form and take logs: 2 = ekt, so t = ln 2 / k. Set A/A₀ = ½ instead and you get t = ln 2 / |k| with k negative. Same logarithm, same divisor, one growing and one shrinking. The number 0.693147 turns up everywhere in this subject for that reason and no other — it is ln 2, and nothing more.
Worked example: 1,000 units at 5% per period for ten periods
Start with A₀ = 1,000, a rate of 5% per period, and t = 10 periods.
- Growth factor for one period. 1 + 0.05 = 1.05.
- Continuous rate. k = ln(1.05) = 0.0487902 per period.
- Growth factor over ten periods. 1.0510 = 1.6288946. Check it with the continuous form: e0.0487902 × 10 = e0.487902 = 1.6288946. The two agree, as they must.
- Final amount. 1,000 × 1.6288946 = 1,628.89.
- Change. 1,628.89 − 1,000 = 628.89, a 62.89% total increase over ten periods.
- Doubling time. ln 2 ÷ 0.0487902 = 0.6931472 ÷ 0.0487902 = 14.2067 periods.
Sanity-check the doubling time against the result: after 14.2067 periods the multiple should be 2, and after 10 periods it is 1.6289, which is less than 2 as it must be since 10 < 14.2067. The rule of 72 would have estimated 72 ÷ 5 = 14.4 periods, which is 0.19 periods high — close enough for mental arithmetic and not close enough for a spreadsheet.
Running it backwards. Enter t = −10 with the same rate and you get 1,000 ÷ 1.6288946 = 613.91: the amount that would have grown to 1,000 over the preceding ten periods. This is the same operation as discounting a future cash flow to the present, which is why the compound interest calculator and this one share their arithmetic.
A decay example. A sample decaying at a continuous rate of k = −0.0693147 per year has a half-life of 0.693147 ÷ 0.0693147 = 10 years exactly. After 10 years, 50% remains; after 20 years, 25%; after 30 years, 12.5%. Each half-life removes half of what is left, never half of the original.
Reading the result, and knowing when the model stops applying
The doubling time is the most transferable number on the page. A rate of 5% per period means little on its own; “doubles every 14.2 periods” is immediately comparable against anything else. It is also scale-free: the doubling time does not depend on the starting amount at all, which is why it is the right figure to quote when comparing two growing systems of very different sizes.
The rule of 72 is a mental approximation with a known error. Divide 72 by the percentage rate for an estimate of the doubling time. It works because ln 2 × 100 = 69.31, and 72 is chosen instead because it has more divisors and because the discrete-to-continuous correction pushes the exact answer upward at ordinary rates. Compare the columns in the table below: the rule sits above the exact figure at low rates and below it at high ones, crossing over between 7% and 8%.
Exponential models describe a regime, not a system. Every real growth process leaves the exponential regime eventually, because the resource that feeds it runs out. Populations hit carrying capacity and follow a logistic curve; adoption curves saturate; compound returns meet drawdowns. Extrapolating an exponential fit far past its data is the single most common misuse of this formula, and the calculator warns you when the projected change exceeds twelve orders of magnitude for exactly that reason.
Decay has the opposite failure mode. The model says a decaying quantity never reaches zero, which is true of a continuous variable and false of a countable one. A sample of a thousand atoms is not described by an exponential once a handful remain, because at that point the process is visibly discrete and random. The half-life is a statistical property of a large population, not a promise about any individual.
Check that your rate and time use the same period. This is the most frequent arithmetic error in exponential work. A rate quoted per year with a time in months gives an answer wrong by a factor you will not spot, because the wrong answer is still smooth and plausible. Convert one of them first.
Doubling time by growth rate, exact and by the rule of 72
| Rate per period | Continuous rate k | Exact doubling time | Rule of 72 estimate |
|---|---|---|---|
| 1% | 0.0099503 | 69.6607 | 72.0000 |
| 2% | 0.0198026 | 35.0028 | 36.0000 |
| 3% | 0.0295588 | 23.4498 | 24.0000 |
| 4% | 0.0392207 | 17.6730 | 18.0000 |
| 5% | 0.0487902 | 14.2067 | 14.4000 |
| 6% | 0.0582689 | 11.8957 | 12.0000 |
| 7% | 0.0676586 | 10.2448 | 10.2857 |
| 8% | 0.0769610 | 9.0065 | 9.0000 |
| 10% | 0.0953102 | 7.2725 | 7.2000 |
| 12% | 0.1133287 | 6.1163 | 6.0000 |
For decay, read the same rows with a negative rate: a loss of 5% per period has a half-life of 13.5135 periods, not 14.2067, because ln(0.95) = −0.0512933 is larger in magnitude than ln(1.05).
Mistakes that produce a plausible wrong answer
- Mismatched period units. A rate per year with a time in months is the commonest error here, and the result is smooth and believable rather than obviously broken.
- Treating a percentage rate as a continuous rate. Entering 5 as k means k = 0.05 per period, which is a per-period growth factor of e0.05 = 1.05127, not 1.05. Small at one period, material over fifty.
- Assuming decay at r% has the same time constant as growth at r%. It does not, because ln(1 + r) and ln(1 − r) differ in magnitude. At 5%, doubling takes 14.21 periods but halving takes 13.51.
- Adding growth rates across periods. Two consecutive periods at 10% is a factor of 1.21, not 1.20. Multiply the factors; never add the rates.
- Extrapolating far beyond the data. An exponential fitted to five points will happily predict a number larger than the observable universe. The fit is not evidence that the regime continues.
- Reading a half-life as a lifetime. After one half-life, half remains; after two, a quarter. The mean lifetime is 1/|k|, which is 1/ln2 = 1.4427 times the half-life, not twice it.
Where this model sits among its relatives
Geometric sequences are the discrete case. Sampling A₀(1 + r)t at whole-number times gives exactly a geometric progression with common ratio 1 + r, so the geometric sequence calculator answers the same questions when time comes in whole steps and you also want the sum of the terms. Its arithmetic counterpart, the arithmetic sequence, is the linear model this one is usually contrasted with.
Compound interest is this formula with financial vocabulary. Principal is A₀, the annual rate is r, and compounding more often than once a period means replacing (1 + r)t with (1 + r/n)nt, which converges on ert as n grows. That limit is where continuous compounding comes from, and it is the reason e appears in finance at all.
Logarithms are the inverse operation. Any question of the form “how long until the quantity reaches X” is solved by taking a logarithm: t = ln(X/A₀) / k. The logarithm calculator handles the general case, including changing base, and the exponent calculator covers the forward direction with arbitrary bases.
Where the exponential breaks, the logistic takes over. When growth is limited by a carrying capacity K, the model becomes dA/dt = kA(1 − A/K), which behaves exponentially while A is small compared with K and flattens as A approaches it. The early portion of a logistic curve is indistinguishable from an exponential, which is precisely why exponential fits to early data are so persuasive and so often wrong about the endgame.
For very large or very small results, express them in scientific notation: a quantity that has grown by fifteen doublings is 215 = 32,768 times its start, and the digits stop being readable long before the exponent stops being meaningful.