Population Doubling Time Calculator
A small steady growth rate can hide how fast something really expands, and doubling time makes it vivid by answering one question: how long until the population is twice as large? For a quantity growing at a constant continuous rate, the doubling time is simply the natural logarithm of 2 divided by that rate. The logic is direct: doubling means e raised to the rate times the time equals 2, and taking the natural log of both sides gives rate times time equals ln(2), so the time is ln(2) divided by the rate. With a growth rate of 3.5 percent per year, the doubling time is 0.693147 divided by 0.035, about 19.80 years. This is the precise continuous form of the familiar rule of 70, where you divide 70 by the percentage rate for a quick estimate, and it works because ln(2) is close to 0.70. The same formula applies to anything that compounds continuously, from bacteria to investments to energy demand, with only the meaning of the rate changing. Enter a growth rate and the page returns the doubling time and the equivalent rule-of-70 estimate. Every figure is computed deterministically from the formula, never estimated, with the method and a worked example shown below for verification.
Doubling time is the natural log of 2 over the growth rate: t = ln(2) / r. A population growing at 3.5% per year doubles in about 19.80 years.
Doubling time formula
doubling time = ln(2) / r
ln(2) = 0.693147...
r = continuous growth rate per period (as a decimal)
rule of 70: doubling time is about 70 / rate in percent
Doubling requires e^(r x t) to equal 2. Taking the natural log gives r times t equals ln(2), so dividing ln(2) by r yields the time to double.
Worked example
A population grows at 3.5 percent per year.
- r = 3.5 / 100 = 0.035
- ln(2) = 0.693147
- Doubling time = 0.693147 / 0.035 = 19.8042
- Rule of 70 check: 70 / 3.5 = 20 years, close to the exact value
The population doubles in about 19.80 years. This is the calculator's default input, so the result above matches the widget exactly.
Doubling time by rate
Exact continuous doubling time using ln(2) divided by the rate.
| Rate | Doubling time |
|---|---|
| 1% | 69.31 periods |
| 2% | 34.66 periods |
| 3.5% | 19.80 periods |
| 7% | 9.90 periods |
Population ecology and growth rates: US Geological Survey (USGS).
Population doubling time calculator: frequently asked questions
How do you calculate doubling time?
For continuous exponential growth, doubling time is the natural logarithm of 2 divided by the growth rate. With a rate of 3.5 percent per year, doubling time is 0.693147 divided by 0.035, which is about 19.80 years.
Why use the natural log of 2?
Doubling means the population reaches twice its size, so e raised to the rate times the doubling time equals 2. Taking the natural log of both sides gives rate times doubling time equals ln(2), and solving for the time divides ln(2) by the rate.
How does this relate to the rule of 70?
The rule of 70 is a quick approximation: divide 70 by the growth rate in percent. It works because ln(2) is about 0.693, and 0.693 times 100 is close to 70. The exact continuous figure uses ln(2) divided by the decimal rate.
Does this work for any growing quantity?
Yes. The same formula gives the doubling time for anything growing at a constant continuous rate, including investments, bacteria or energy demand. Only the meaning of the rate changes; the math is identical.
Is the result computed automatically?
Yes. The page divides ln(2) by the growth rate deterministically. No value is estimated or hard-coded, so changing the rate updates the doubling time instantly.
Official sources
- Population ecology and growth rates: US Geological Survey (USGS). As at 25 June 2026.
Reviewed by the CalculatorHub team, edited by James Graham, 25 June 2026. See our methodology. This is general information, not financial, tax, legal or investment advice.