exponentialgrowthcalculator.com

Doubling Time Calculator

Enter a growth rate to see the exact doubling time and the quick Rule-of-70 estimate side by side, with the gap between them updating live.

Doubling time calculator

%
Percent increase per period
Exact Doubling Time 11.90 periods at 6% growth
Rule-of-70 Estimate 11.55 0.693 ÷ r
Approximation Error 0.35 periods the estimate runs short

Exact vs. Rule-of-70 Across Rates

Rate Exact t₂ Rule-of-70 t₂ Difference Difference as % of Exact
1% 69.66 69.30 0.36 0.52%
2% 35.00 34.65 0.35 1.01%
5% 14.21 13.86 0.35 2.44%
10% 7.27 6.93 0.34 4.71%
15% 4.96 4.62 0.34 6.85%
20% 3.80 3.46 0.34 8.86%

What the table shows: the raw gap between the two methods, measured in periods, barely moves, 0.36 periods at a 1% rate versus 0.34 periods at 20%. What changes is how big that gap looks next to the answer itself: at 1% it is only 0.52% of the exact doubling time, but at 20% the same roughly one-third-of-a-period gap has grown to 8.86% of a much shorter answer. The Rule-of-70 shortcut does not get numerically worse so much as the true doubling time shrinks faster than its error does.

Definition

What Is Doubling Time?

Doubling time is the length of time a quantity growing at a constant percentage rate takes to become exactly twice its current size, and that length of time never depends on the value it started from.

A quantity that doubles in 10 years takes another 10 years to double again from that new, larger level, and another 10 after that. The rate never changes, so the time needed to double never changes either, only the absolute amount added during each doubling grows, because 100% of a bigger number is a bigger number.

This independence from starting value falls straight out of the algebra. Doubling time solves x₀(1 + r)ᵗ = 2x₀ for t, and x₀ appears on both sides of that equation in exactly the same way, so it cancels out before t is ever isolated. Whatever the starting quantity was, it plays no role in how long doubling takes, only the rate r does.

It is also constant throughout the entire growth process, not just at the start. Measured from any point along the curve, not only from t = 0, the time to the next doubling is identical, because the growth factor applied each period never changes. This is the property that makes doubling time useful as a single, memorable number instead of a curve that has to be read off a graph.

Equation

The Doubling Time Formula

The doubling time formula is t₂ = ln(2) / ln(1 + r) for the exact discrete answer, or the faster t₂ ≈ 0.693 / r for the Rule-of-70 approximation. Both take a periodic growth rate and return how many periods it takes to double.

Exact, solved from the growth equation
t2 = ln(2) / ln(1 + r)

Derived by setting the growth formula equal to 2x₀ and solving for t. Correct at any rate, however large.

Rule of 70, mental-math shortcut
t2 ≈ 0.693 / r

Replaces ln(1 + r) with r itself, which is a close stand-in only while r stays small. Fast to compute without a calculator.

Variables in the doubling time equations
SymbolNameWhat it representsExample
t2Doubling timePeriods needed for the quantity to reach exactly twice its current value.11.90 periods
rPeriodic growth rateThe proportional increase per period, as a decimal.0.06 (that is 6%)
ln(2)Natural log of 2A fixed constant, roughly 0.6931, from solving (1 + r)ᵗ = 2.0.6931
0.693Rounded ln(2)The numerator used in the Rule-of-70 shortcut in place of the full log calculation.0.693

What the table shows: every quantity in either formula reduces to just two inputs a user actually supplies, the rate r and, implicitly, the constant ln(2). t2 is always the computed result, never something entered directly.

Method

How to Calculate Doubling Time

To calculate doubling time, convert the growth rate to a decimal, add 1 to get the growth factor, take the natural log of that factor, then divide the natural log of 2 by the result.

  1. Convert the growth rate to a decimal

    Divide the percentage rate by 100. A rate of 4% per period becomes r = 0.04.

  2. Add 1 to get the growth factor

    Calculate 1 + r. With r = 0.04 the growth factor is 1.04.

  3. Take the natural log of the growth factor

    Compute ln(1.04). This equals approximately 0.0392.

  4. Divide ln(2) by that value

    ln(2) ≈ 0.6931. Dividing gives 0.6931 / 0.0392 ≈ 17.67, the exact doubling time.

  5. For a fast estimate, skip the log entirely

    Divide 0.693 by the decimal rate directly: 0.693 / 0.04 = 17.33. Compare it against step 4 to see how far the shortcut has drifted.

Worked example, a growing app's weekly active users

A mobile app has 1,200 weekly active users and is growing 4% per week. Both formulas answer the same question: how many weeks until the app reaches 2,400 users?

x0
1,200 users
r
4% → 0.04
Target
2,400 users (2×)
t2 = ln(2) / ln(1 + r) t2 = ln(2) / ln(1.04) t2 = 0.6931 / 0.0392 t2 = 17.67 weeks (exact)

The Rule-of-70 shortcut gives 0.693 / 0.04 = 17.33 weeks, predicting 2,400 users about 2.4 days earlier than the exact crossover at week 17.67. For weekly planning that gap is small enough to ignore; for a precise launch-date forecast it is not.

Accuracy

Why the Rule-of-70 Approximation Drifts at Higher Rates

The Rule-of-70 shortcut drifts at higher rates because it replaces ln(1 + r) with r itself, and that substitution only holds well when r is small. The natural log function ln(1 + r) can be expanded as r − r²/2 + r³/3 − ..., which is almost exactly r when r is tiny but pulls further below r as r grows, because the r²/2 term and beyond stop being negligible.

Since the approximation always divides 0.693 by a number larger than the true ln(1 + r) it should be using, the estimate always comes out a little short. It never overshoots. At a 1% growth rate the shortfall is under 1% of the true answer, small enough to ignore for almost any purpose. By a 20% growth rate, the same style of gap has grown to nearly 9% of the true doubling time, which starts to matter for anything more precise than a rough mental estimate.

The practical rule of thumb: treat the Rule-of-70 shortcut as reliable for growth rates under roughly 10%, the range where compound interest, population growth, and most business metrics usually live. Beyond that, or whenever the exact figure actually matters, use t₂ = ln(2) / ln(1 + r) instead, the same log-based approach used by the calculator across this whole site. For growth modeled continuously rather than in discrete periods, the continuous-rate version of this page uses the related, even simpler t₂ = ln(2) / k, with no approximation needed at all.

Questions

Frequently Asked Questions

What is doubling time?

Doubling time is the length of time a quantity growing at a constant percentage rate takes to become exactly twice its current size. A city growing 3% a year, an investment compounding at 7% annually, and a bacterial colony splitting every generation are all governed by the same doubling-time math, only the rate changes.

Because the growth rate stays fixed, doubling time stays fixed too: it takes the same span to go from 1,000 to 2,000 as it does to go from 1,000,000 to 2,000,000.

What is the exact formula for doubling time?

The exact discrete doubling time is t₂ = ln(2) / ln(1 + r), where r is the periodic growth rate written as a decimal. This comes directly from solving x₀(1 + r)ᵗ = 2x₀ for t. The initial value x₀ cancels out of both sides, which is also the proof that doubling time never depends on where the growth started.

At a 6% periodic growth rate this gives ln(2) / ln(1.06) = 0.6931 / 0.05827 ≈ 11.90 periods.

What is the Rule-of-70 shortcut, and how accurate is it?

The Rule-of-70 shortcut estimates doubling time as t₂ ≈ 0.693 / r, replacing the exact denominator ln(1 + r) with the plain rate r. It is accurate to within a few hundredths of a period for small rates, under about 5%, but understates the true doubling time by an increasing share of the answer as the rate climbs, because ln(1 + r) drifts further below r the larger r gets.

The calculator above shows both numbers at once specifically so the size of that gap is never hidden.

Does doubling time depend on the starting value?

No, doubling time depends only on the growth rate, never on the starting value. Solving x₀(1 + r)ᵗ = 2x₀ for t cancels x₀ from both sides algebraically, leaving t₂ = ln(2) / ln(1 + r) with no trace of the initial quantity.

That is why a 6% growth rate gives the same 11.90-period doubling time whether the starting quantity is 10 units or 10 million.

How does doubling time relate to half-life?

Doubling time and half-life are the same calculation run in opposite directions: t₂ = ln(2) / ln(1 + r) for a growing quantity, and t½ = ln(2) / (−ln(1 − r)) for a shrinking one. Both answer "how long until this changes by a factor of two", one asks about multiplying by two, the other about dividing by two.

The dedicated half-life calculator runs the shrinking-quantity version directly.

Why is the shortcut sometimes called the Rule of 72 instead of 70?

The Rule of 72 swaps 0.693 for 0.72 (equivalently 72 for the percentage form) because 72 divides evenly by more small numbers, 2, 3, 4, 6, 8, 9, 12, making it easier for mental arithmetic than the more mathematically exact 69.3. Both are approximations of the same ln(2) ≈ 0.6931 constant; 72 trades a little accuracy for easier division by hand.

This page uses the more precise 0.693 form throughout so the comparison against the exact value is as fair as possible.

What real-world situations use doubling time?

Doubling time is commonly used for compound investment growth, population growth, early-stage epidemic case counts, and computing capacity such as Moore's Law transistor counts. In each case a quantity is changing by a roughly constant percentage per period, which is exactly the condition doubling time measures.

It is also used in reverse, informally, to describe how fast prices erode purchasing power under a given inflation rate.