One rate, and the four numbers that are all the same fact.
Growth, decay, doubling and half-life
All four are the same fact.
1,000 × (1 + 7%)^10
1,967.151357
Up 967.151357 over 10 years — a total multiplier of 1.96715.
Final amount
1,967.151357
after every period
Doubling time
10.245
years to double
Continuous rate
6.7659%
the same curve as e^rt
Total multiplier
1.96715
over the whole span
First year
70
change in that period
Last year
128.692145
same rate, different amount
Along the way
The amount at intervals across the span, with the change since the previous row
Year
Amount
Change since last row
0
1,000
—
1
1,070
70
2
1,144.9
74.9
3
1,225.043
80.143
4
1,310.79601
85.75301
5
1,402.551731
91.755721
6
1,500.730352
98.178621
7
1,605.781476
105.051125
8
1,718.18618
112.404703
9
1,838.459212
120.273033
10
1,967.151357
128.692145
The change column grows (or shrinks) row by row even though the rate never moves. That widening gap is the whole difference between exponential and linear, and it is why a projection that looks gentle early can be startling late.
7% per year over 10 years multiplies the starting amount by 1.96715, not by 1.70000 — the difference between compounding and adding the same percentage each time.
At this rate the quantity doubles every 10.245 years. The "rule of 70" estimates that as 70 ÷ 7 = 10.00, which is close because ln 2 is about 0.693.
The first year adds 70.0000 and the last adds 128.692. Same percentage, different amount — that gap IS what exponential means.
Expressed continuously the same curve is 6.765865% per year, which is what an e^rt model would use. It differs from the periodic rate for the same reason an APR differs from an APY.
Nothing here checks whether the quantity really grows exponentially. Populations meet limits, adoption curves saturate, and a smooth projection is an assumption you are making rather than a finding.
Doubling time, half-life and the continuous rate are all derived from the same rate, so they cannot contradict each other.
What this tool shows
7% a year doubles in 10.24 years. A 7% decline halves in 9.55 — sooner, not later, because losing 7% is a bigger proportional step than gaining it. Growth rate, doubling time, half-life and the continuous rate are one quantity written four ways.
A quantity projected at a constant percentage rate
Doubling time, or half-life for a decline
The equivalent continuous rate
The total multiplier over the whole span
The change in the first period against the last
The curve tabulated along the way
Growth and decay Doubling and half-life The continuous rate The curve tabulated
Updated 7 September 2026 · Works in any browser, no installation
final = initial × (1 + r)n. 1,000 at 7% for 10 years is 1,000 × 1.07¹⁰ = 1,967.15. The rate applies to the NEW amount each period, which is why the answer is not 1,000 + 10×70 = 1,700.
At a glance
Formula shown
final = initial \u00d7 (1 + r)\u207f. Doubling time is ln 2 \u00f7 ln(1 + r); half-life is ln 0.5 \u00f7 ln(1 + r). The equivalent continuous rate is ln(1 + r), which is what an e^rt model uses.
Scenario support
Projecting a balance, a population or a user base; working out a half-life from a decay rate; checking how long a quantity takes to double.
Educational estimate
Planning support from the values you enter — not professional advice.
What exponential actually means
A constant percentage is not a constant amount. That is the whole idea, and it is worth seeing in numbers.
1,000 growing at 7%: the first year adds 70. The tenth year adds 128. The rate never changed; the base it applies to did.
Linear growth adds the same amount every period, so ten years at +70 gives 1,700. Exponential growth gives 1,967. The gap widens without limit as the span grows, which is why long projections are so sensitive to the rate.
The table on this page shows the period-on-period change growing row by row. That column is the difference between exponential and linear, made visible.
Doubling time and the rule of 70
Doubling time depends only on the rate. A quantity growing at 7% doubles every 10.24 periods whatever it started at.
Exactly: ln 2 ÷ ln(1.07) = 10.2448. The familiar shortcut is the rule of 70 — 70 ÷ 7 = 10 — which works because ln 2 is about 0.693 and, for small rates, ln(1 + r) is close to r.
The approximation is good to within a few per cent for rates up to about 10% and drifts after that. At 20% the rule gives 3.5 and the true answer is 3.8. The page gives both, so the shortcut can be checked rather than trusted.
The asymmetry that surprises
A rise of 7% doubles in 10.24 periods. A fall of 7% halves in 9.55. Most people expect those to match, and they do not.
The reason is that a 7% loss is a bigger proportional step than a 7% gain. In log terms, |ln 0.93| = 0.0726 while ln 1.07 = 0.0677 — so the decline covers ground faster, and it reaches the halfway point sooner.
It is the same fact behind a familiar one: lose 50% and you need a 100% gain to get back. The two directions are not mirror images, and treating them as such is a real modelling error.
Worth being honest about: an earlier draft of this page said the opposite — that a decline takes longer. The test suite raised the growth factor to the reported half-life and checked what came back, which caught the prose as well as the arithmetic.
Periodic and continuous rates
The same curve can be described two ways, and the two rates are not the same number.
Periodic: multiply by (1 + r) once per period. This is how interest is normally quoted and how this page takes its input.
Continuous: multiply by ert, compounding at every instant. The continuous rate matching 7% periodic is ln(1.07) = 6.766%.
Both produce identical values at whole periods; they differ in how the rate is stated. It is the same relationship as between an APR and an APY, and it is why comparing a continuous rate with a periodic one directly overstates the difference.
Physics and biology usually work continuously; finance and ordinary reporting usually work periodically. The page gives both so a figure from one world can be read in the other.
Where it genuinely applies
Compound interest. The definitional case. A balance at a fixed rate is exactly this curve.
Radioactive decay. Genuinely exponential, and the reason half-life is a meaningful constant rather than an approximation.
Early epidemic spread. Exponential while susceptible people are plentiful, and it stops being so as soon as they are not — which is the point of the model, not a failure of it.
Drug clearance. First-order elimination is exponential, which is why dosing intervals are set from half-lives.
Cooling and discharge. A body cooling towards ambient, or a capacitor discharging, both follow an exponential approach to a limit.
Where it does not
Nothing grows exponentially forever, and this page does not check whether yours does.
Populations meet limits. Food, space and disease bend the curve. Logistic growth — exponential early, flattening towards a ceiling — is the model demographers actually use.
Adoption saturates. Users, market share, installed base: all of them run out of people to convert.
Rates change. Twenty years of “7% a year” assumes twenty years of the same conditions, which is a large assumption stated as a small one.
A projection that continues past where the assumption holds is not a forecast; it is arithmetic wearing a forecast’s clothes. The number will be exact and the answer will still be wrong.
Sources and methodology
The model is standard; the references are for the conventions and for where it is known to break down.
Method. The projection is a single power rather than a loop, and the suite checks it against a plain multiplication loop across two thousand generated cases. Doubling time and half-life are verified by raising the growth factor back to them and requiring exactly 2 and 0.5. Solving for any of the four quantities — rate, periods, starting amount, final amount — round-trips through the others, which the suite asserts on two thousand more. That engine is verified on every change against 81 hand-written assertions, including that raising the growth factor to the reported doubling time gives exactly 2, and that losing r% halves sooner than gaining r% doubles at every rate from 1% to 60% — a claim this file originally had backwards. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
Geometric SequenceNth term, partial sum, and whether the infinite series converges at all — with exact ratios, so 1/3 stays 1/3 instead of becoming 0.3333333333.
Doubling TimeHow long a quantity takes to double at a constant rate, or from two measurements — with the rule of 70 and rule of 72 measured against the exact answer.
LogLogarithms in any base with the exponential form beside them, whole answers confirmed by raising the base back, and change of base worked through.
ExponentPowers with the awkward cases right — a negative exponent is a reciprocal not a sign, a fractional one is a root, and zero to the zero is reported as contested.
Geometric MeanThe average for things that compound. Growth of +50% then -50% averages to zero arithmetically and to a real 13.4% loss geometrically, which is what actually happened.
Percent to GoalProgress against a target, plus the pace the remaining periods actually need — the number a progress bar never shows and nobody works out in their head.
Read as a sequence of terms rather than a projection, the same curve is the Geometric Sequence Calculator, which also answers whether an infinite series of them has a total.
Educational use disclaimer
This is an educational tool. Nothing here checks whether the quantity really grows exponentially — populations meet limits and adoption curves saturate, so a smooth projection is an assumption you are making rather than a finding.
Published the exponential growth page showing the growth rate, doubling time, half-life and continuous rate together, because they are one quantity in four costumes.
States the asymmetry that catches people: a 7% decline halves SOONER than a 7% rise doubles, since losing 7% is a bigger proportional step than gaining it. An earlier draft claimed the opposite and the test suite caught it.
Says plainly that nothing checks whether the quantity really grows exponentially — populations meet limits and adoption saturates, so a smooth projection is an assumption rather than a finding.
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