Doubling Constant Calculator
How long until it doubles? Enter a growth rate and this doubling constant calculator gives the exact doubling time from ln(2), the constant 0.693147 behind every doubling calculation, side by side with the rule of 70 and rule of 72 estimates and the error in each. Or flip the tabs: enter a target time and find the growth rate needed to double in it.
The tool works in periods, so the same maths covers yearly investment returns, monthly growth, inflation, population growth and bacterial cultures. Pick periodic or continuous compounding to match how your quantity actually grows.
Quick Answer
Key takeaways
- The doubling constant is ln(2), about 0.693147. Every exact doubling formula is that constant divided by a growth term.
- Periodic growth uses t = ln(2) / ln(1 + r); continuous growth uses t = ln(2) / r and always doubles a little faster.
- The rule of 72 is a shortcut, not the answer: this tool prints its error, and the error grows past 3% once rates pass the mid teens.
- The reverse mode answers the planning question directly: name the doubling time, get the rate it demands.
- Quick Answer
- How to use the doubling constant calculator
- The doubling constant: where ln(2) comes from
- Worked example: 8% growth, checked three ways
- Doubling time by growth rate: exact answers vs both rules
- Why 72 and not 69.3
- Where doubling time shows up outside investing
- What competing calculators miss
- Frequently Asked Questions
- Add This Calculator to Your Website
How to use the doubling constant calculator
Pick a direction first. Rate to doubling time is the default: enter a growth rate per period, choose periodic or continuous compounding, and read the exact answer with both rules and their errors underneath. Time to required rate runs the maths backwards: enter the doubling time you want and the tool solves for the rate, which is the mode to use for savings targets and growth goals. If your rate comes from comparing two periods of growth rather than a stated return, a compound annual growth calculator will derive that smoothed yearly rate first.
Periods are whatever your rate is measured in. An 8% yearly return gives an answer in years; a 2% monthly growth rate gives an answer in months. The tool never assumes years, which is why its results say periods.
The doubling constant: where ln(2) comes from
The doubling constant is ln(2) because doubling means solving for the moment a growing quantity reaches twice its start, and the natural logarithm of 2 is what falls out of that equation.
Continuous growth: start from e^(rt) = 2, and t = ln(2) / r.
Reverse, periodic: r = (2^(1/t) – 1) x 100. Reverse, continuous: r = ln(2) / t x 100.
ln(2) = 0.693147 to six places. Textbooks state the same result: OpenStax Calculus Volume 2, in its section on exponential growth and decay, defines doubling time as ln(2) divided by the growth constant, and OpenStax College Algebra, in its exponential and logarithmic models section, derives t = ln(2) / k from 2A0 = A0 e^(kt). The formula only holds while the rate stays constant, the same assumption behind a constant growth rate calculation
One naming note: in reactor physics, doubling time also appears in neutron kinetics, where a period is the time for power to rise by a factor of e. This calculator uses the general exponential growth meaning, the one used in finance, demographics and biology, and its constant is ln(2) in every mode.
Worked example: 8% growth, checked three ways
Enter 8 with periodic compounding and the tool returns an exact doubling time of 9.006 periods, verified against hand calculation: ln(2) / ln(1.08) = 0.693147 / 0.076961 = 9.006.
Switch compounding to continuous and the exact answer drops to 8.664 periods, since growth at every instant outruns growth once a period: ln(2) / 0.08 = 8.664. Then flip to the reverse tab, enter 10, and the tool asks for 7.177% per period, from (2^(1/10) – 1) x 100. If you already hold a balance and want the date it doubles under regular deposits rather than a bare rate, a backward compound interest calculator works the same logarithm from the balance side.
Doubling time by growth rate: exact answers vs both rules
Every row below is output from this calculator with periodic compounding. The pattern matters more than any single row: the rule of 72 crosses the exact answer near 8%, overstates doubling time below it and understates it above it, while the rule of 70 tracks continuous growth better than periodic.
| Growth rate | Exact (periods) | Rule of 72 | Rule of 70 |
|---|---|---|---|
| 2% | 35.003 | 36.000 | 35.000 |
| 3% | 23.450 | 24.000 | 23.333 |
| 4% | 17.673 | 18.000 | 17.500 |
| 5% | 14.207 | 14.400 | 14.000 |
| 6% | 11.896 | 12.000 | 11.667 |
| 8% | 9.006 | 9.000 | 8.750 |
| 10% | 7.273 | 7.200 | 7.000 |
| 12% | 6.116 | 6.000 | 5.833 |
| 20% | 3.802 | 3.600 | 3.500 |
At 2% the rule of 72 is a full period out, which is a year of planning error on a savings goal. At 20% both rules understate the wait by more than 5%. The exact formula costs nothing extra here, so the rules are best kept for mental estimates away from a screen, the use the Investopedia definition of the rule of 72 describes: a quick gauge, with calculators doing the precise work.
Why 72 and not 69.3
The most accurate shortcut numerator would be 69.3, since 100 x ln(2) = 69.31. Two things keep 72 in use. First, 72 divides evenly by 2, 3, 4, 6, 8, 9 and 12, so the mental division is trivial at the rates people actually meet. Second, periodic compounding bends the exact answer upward relative to the continuous formula, and the slightly larger numerator roughly cancels that bend between about 6% and 10%, which is why the rule of 72 error at 8% is only -0.07%. The rule of 70 sits between the two: less divisible, but closer for continuous growth, where at 8% it errs by +0.99% against the rule of 72 at +3.87%. For the compounding effect itself, the SEC Investor.gov guide to the power of compounding walks through how returns earning their own returns is what makes the doubling schedule constant in the first place.
Where doubling time shows up outside investing
Anything growing at a steady percentage rate doubles on this schedule. Prices under 3% inflation double in 23.450 years, which is the same calculation as money doubling at 3%, run against your purchasing power. A city growing at 2% a year doubles its population in 35.003 years. A bacterial culture growing at a steady rate doubles on the period the biology texts measure directly, and dividend income reinvested at a steady yield follows the same curve, which a drip compound calculator projects balance by balance. Moore’s law was simply a doubling time claim, roughly two years per doubling, stated about transistor counts.
What competing calculators miss
Most doubling time tools give one number. One popular version adds contexts and charts but reports only the rule of 72 beside the exact answer, with no rule of 70 and no printed error, so you cannot tell whether the shortcut is safe at your rate. Another widely used page handles the exact formula well and even discusses negative rates, but it has no reverse mode, so the planning question, what rate do I need, goes unanswered. Rule of 72 pages in general skip the exact logarithm entirely. This calculator keeps all four pieces together: the exact answer, both rules with signed errors, continuous as well as periodic compounding, and the reverse solver. A close cousin worth knowing about is the compound interest doubling time calculator, which ties the same logarithm to a starting balance and interest inputs; this page is the general one, for any quantity growing at any steady rate, money or not.
Frequently Asked Questions
The doubling constant is ln(2), approximately 0.693147. It is the numerator in every exact doubling time formula: divide it by the growth rate (continuous) or by ln(1 + r) (periodic) to get the time a quantity needs to double.
For growth once per period, t = ln(2) / ln(1 + r), where r is the rate per period as a decimal. For continuous growth, t = ln(2) / r. At 8% per year with yearly growth, t = 9.006 years.
It depends on the rate and the compounding. At 8% periodic growth the rule of 72 gives 9.000 against an exact 9.006, an error of -0.07%, while the rule of 70 gives 8.750, an error of -2.85%. For continuous growth the rule of 70 is usually closer, which is why this calculator shows both errors.
Exactly 10.245 years with yearly compounding. The rule of 72 estimates 10.286 years and the rule of 70 estimates 10.000 years.
7.177% per period with periodic compounding, or 6.931% with continuous compounding. Use the Time to required rate mode and enter 10 as the doubling time.
69.3 is 100 times ln(2), so it is the most accurate numerator for continuous growth. 72 is close at typical periodic rates and divides evenly by 2, 3, 4, 6, 8, 9 and 12, which makes the mental maths far easier.
Yes. Any quantity growing at a constant percentage rate doubles on the same schedule, so at 3% inflation prices double in 23.450 years, and a population growing at 2% a year doubles in 35.003 years.
Poorly. At 20% the exact doubling time is 3.802 periods, but the rule of 72 says 3.600, an error of -5.31%. The rules are shortcuts for moderate rates; use the exact formula when the rate is high or the answer matters.
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References
- OpenStax, Rice University, Calculus Volume 2, Section 2.8: Exponential Growth and Decay, defining doubling time as ln(2) divided by the growth constant and deriving continuous compounding. OpenStax: Calculus Volume 2, exponential growth and decay. Accessed October 2026.
- OpenStax, Rice University, College Algebra with Corequisite Support, Section 6.7: Exponential and Logarithmic Models, deriving the doubling time formula t = ln(2) / k. OpenStax: College Algebra, exponential and logarithmic models. Accessed October 2026.
- OpenStax, Rice University, Contemporary Mathematics, Section 3.11: Geometric Sequences, on yearly compounding A = P(1 + r)^t, the periodic model behind t = ln(2) / ln(1 + r). OpenStax: Contemporary Mathematics, geometric sequences. Accessed October 2026.
- Investopedia, “Rule of 72 Definition, Usefulness, and How to Use It”, on the rule as a quick estimate of the years needed to double invested money at a given annual rate of return. Investopedia: Rule of 72 definition and how to use it. Accessed October 2026.
- U.S. Securities and Exchange Commission, Investor.gov, “It’s Tax Time: Getting a Tax Refund? Consider Investing It”, on the power of compounding and earning returns on past returns. Investor.gov (SEC): The power of compounding. Accessed October 2026.
- NerdWallet, “Rule of 72: What It Is and How It Works”, on using the rule to estimate doubling time and its accuracy range. NerdWallet: Rule of 72, what it is and how it works. Accessed October 2026.