Doubling Time Calculator

Why doubling time is a logarithm and not a division

Growth at a constant rate multiplies rather than adds, and that is the whole reason a logarithm appears. A quantity growing 7% a year is multiplied by 1.07 every year, so after t years it stands at 1.07^t times where it began. Asking when it doubles means asking which power of 1.07 equals 2 — and pulling an exponent down out of a power is exactly what a logarithm does. Taking logs of both sides of 1.07^t = 2 gives t × ln(1.07) = ln(2), and dividing leaves the formula above.

The base does not matter, which is worth knowing because it means the identity can be checked with whatever log key is nearest: log₂(2) ÷ log₂(1.07) gives 10.24 just as ln does, since the change-of-base factor cancels top and bottom. What matters is that the same fixed number, ln 2 = 0.693147, sits in the numerator of every doubling time ever calculated. Everything specific to your situation is in the denominator.

One consequence follows immediately and is the most useful thing on this page: the doubling time does not depend on the starting quantity at all. Two cells and two billion cells growing at the same rate double in the same time. That is why a doubling time can be quoted as a property of a process — a culture, an epidemic, a portfolio’s assumed return — without saying how big the thing currently is.

The rule of 72 is right at about 7.8% and drifts either side of it

Most people meet doubling time as a shortcut: divide 72 by the percentage rate. It is a genuinely good approximation, and it is worth knowing exactly how good, because the answer is “very, in a narrow band”. The rule of 70 is the same idea with a rounder numerator — 70 is closer to 100 × ln 2 = 69.31, which makes it the more accurate of the two at low rates, while 72 divides more neatly and is better around the rates people actually quote. Every row below compares both against the exact formula:

The crossover — the rate at which 72 ÷ r lands exactly on the true answer — is 7.8469%, found by solving the two expressions against each other rather than by rounding to “about 8”. Between roughly 5% and 12% the rule of 72 is within about 1%, which is why it survives in finance, where quoted returns live in that band. Outside it the drift is real: at 50% a year it says 1.44 periods against a true 1.71, and at 1% it is +3.36% too slow.

Neither rule is a different theory of growth — both are the same first-order approximation of the logarithm, with the numerator chosen for arithmetic convenience. Use them for mental estimates, and use the calculator when the number goes into a document.

Two measurements are enough: no rate required

Often no rate is known and there is no need to find one first. Two readings and the time between them contain the doubling time already, because the ratio of the readings is the growth. Divide the second by the first, take log base 2 to count how many doublings that ratio represents, and divide the elapsed time by that count.

A culture read at 2×10⁵ cells and again at 1.6×10⁶ cells 72 hours later has grown eightfold. Eight is 2³, so that is exactly 3 population doublings, and 72 ÷ 3 = 24 hours per doubling. The same two readings imply a growth rate of 2.9302% per hour, which the calculator reports beside the doubling time so the two views of one growth stay together.

The count of doublings need not be a whole number, and usually is not. A reading that rises from 1,200 to 2,400 over 48 hours is precisely 1 doubling, so the doubling time is the whole 48 hours; a rise to 2,000 instead would be 0.7370 of a doubling and a doubling time of 65.13 hours. Nothing about the method changes — the division simply lands on a fraction.

Both measurements have to be above zero. A ratio needs something to divide by, and a logarithm needs a positive argument, so a reading of zero has no doubling time rather than an infinite one. The calculator says so rather than returning a number.

What a doubling time feels like at everyday rates

A few of these are worth carrying around. At 1% a year a quantity takes about seventy years to double — a human lifetime — while at 10% it takes about seven, and at 24% it takes roughly three. The relationship is not linear: halving the rate roughly doubles the time, because the rate sits inside a logarithm.

Compounding is what makes this counter-intuitive. Growing 7% a year for 10.24 years does not add 7 × 10.24 = 71.7%; it adds 100%, because each year’s growth is calculated on a base that last year’s growth already raised. The gap between those two figures is the whole of compound growth, and it widens with every period. At the same 7%, tripling takes 16.24 years and a tenfold rise takes 34.03 — each further multiple costs less extra time than the one before it, which is the same logarithm working in the other direction.

A shrinking quantity has a halving time, not a negative doubling time

Put a negative rate into ln(2) ÷ ln(1 + r) and the arithmetic returns a negative number, because the logarithm of a factor below 1 is negative. That number is not a doubling time and it is not an answer. Something falling never reaches twice its size, so its doubling time does not exist; what exists is the mirror question, and it uses ln(0.5) in the numerator instead.

At −5% a year the halving time is 13.51 years, and at −10% it is 6.58. At exactly −50% it is one period, which is the cleanest check available that the formula is behaving: halving once at a rate that halves is a single step. The calculator returns nothing at all for the doubling time in these cases and relabels the result rather than printing a minus sign, because a negative duration invites being read as a positive one further down a spreadsheet.

Two edges have no answer either way. A rate of exactly zero never doubles and never halves, however long you wait — which is a real fact about it, not a failure of the formula. A rate at or below −100% takes the quantity to zero or through it within a single period, and there is no ongoing rate of decay to extrapolate from; the logarithm of a non-positive growth factor does not exist.

Marker and tumour doubling times: the same arithmetic, a different question

Doubling time is used clinically — for serum markers such as hCG, PSA and calcitonin, for lymphocyte counts, and for the volume of a lung nodule. The arithmetic is identical to the two-measurement mode above: two values, the interval between them, and a logarithm. If you enter a pair of results here you will get the correct number.

The number is not the finding. What a doubling time means for a particular person depends on the assay used and its reference range, the interval between the draws and the assay’s own variability across it, the stage and history it sits in, and the guideline the clinician is working to — several of which use methods other than a two-point calculation, such as a log-slope fitted across three or more results. Those judgements are why this page carries no thresholds, no reference ranges, no interpretation of what any value implies, and no comparison against published cut-offs. Adding them would make it a diagnostic tool, and it is a calculator.

For a volume doubling time from imaging there is a further step this page does not take: nodule reports usually give a diameter, and volume grows as the cube of it, so a diameter measurement must be converted before any doubling time is meaningful. Getting that wrong understates growth by a factor of three, which is exactly the kind of error that should be caught by the person responsible for the result.

Bring any figure you calculate here to the clinician who ordered the test. That is not a formality: it is the only step in the process that can tell you what the number means.

In a spreadsheet, it is one LN and a division

Excel and Google Sheets both have the natural logarithm as LN, so every calculation on this page is a single formula:

• Doubling time from a rate in A1 entered as a real percentage: =LN(2)/LN(1+A1). If A1 holds the plain number 7 rather than 7%, use =LN(2)/LN(1+A1/100).
• Doubling time from two readings with the first in A1, the second in B1, and the elapsed time in C1: =C1/LOG(B1/A1,2). LOG takes its base as the second argument in both applications.
• Halving time for a negative rate in A1: =LN(0.5)/LN(1+A1). Using LN(2) here returns a negative number that looks like a duration and is not one.
• Time to any multiple — triple, ten times, anything — with the multiple in B1: =LN(B1)/LN(1+A1). Setting B1 to 2 reproduces the first formula exactly.

There is nothing to download for any of this and no account to create: the calculator above is a free web page that runs entirely in your browser, on desktop or mobile, and the numbers you type are never sent anywhere.

Read the guide

A doubling time describes a quantity that only compounds — nothing is added to it and nothing taken out. Money almost never behaves that way, and the moment regular contributions enter, a single doubling time stops describing the balance at all: the new money has not been compounding as long as the old, so the total doubles sooner than the rate alone predicts and no one figure covers both. How Compound Interest Works With Regular Contributions works a ten-year example through with monthly deposits and shows where the estimate starts to drift. Read it before applying a doubling time to a savings plan.

Related calculators

Tools that produce the rate this page turns into a time:

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Sources and methodology

The identity itself cites nothing, because there is nothing to cite: ln(2) ÷ ln(1 + r) is derived in full in the first section above rather than asserted. What is cited below is the part that is somebody else’s behaviour rather than mathematics — what a spreadsheet does with these formulas, and the convention by which a cell culture’s growth is expressed in population doublings.

• ATCC — animal cell culture guide (population doubling)
• Microsoft Support — the LN function in Excel
• Google Docs Editors Help — LN (Sheets)
• How we calculate — methodology

Method. Every figure here — the calculator, the step-by-step working, the rule-of-thumb comparison table, and the worked examples in the prose — is produced in your browser by one engine, src/lib/doubling-time.ts, so nothing on the page can disagree with the tool above it. That engine is checked on every change against 88 hand-written assertions, including the cases most likely to be got wrong: that a 100% rate doubles in exactly one period, that a negative rate returns a halving time and no doubling time at all, that two falling measurements do the same, that a rate at or below −100% is refused rather than answered, that the doubling time multiplied by ln(1 + r) always returns ln 2, and that the two-measurement mode and the rate mode agree on the same growth. The crossover rate quoted for the rule of 72 is solved numerically rather than rounded. The count and the per-case breakdown are published on the formula verification page.