Find the exact time for an investment to double, using logarithms instead of the Rule of 72 shortcut.
Solves the compound growth formula exactly for the number of years, using natural logarithms.
Years: ln(2) ÷ ln(1 + rate)
Doubling time is the number of years an investment takes to become twice its size at a steady annual return. This calculator finds it exactly, using logarithms, rather than estimating it with the Rule of 72.
Knowing doubling time turns an abstract percentage into something easy to picture. At 7%, money doubles roughly every ten years; at 12%, about every six. A 30-year-old with 30 working years ahead can then count how many doublings a sum might go through before retirement, which is often the clearest way to see why a few extra points of return, or an earlier start, matter so much.
The same idea applies to prices, population or a company's revenue: anything that grows at a constant percentage rate has a doubling time.
1. Take the annual growth rate R in percent, compounded yearly.
2. Convert it to a growth factor: 1 + R ÷ 100.
3. Take the natural logarithm of 2 (about 0.6931) and of the growth factor.
4. Divide: doubling time = ln 2 ÷ ln(1 + R ÷ 100).
5. Compare with the Rule of 72 estimate, 72 ÷ R, to see how close the shortcut is.
6. To find the rate needed to double in N years, rearrange: R = (2^(1/N) − 1) × 100.
Money doubles when P(1 + r)ⁿ = 2P. The P cancels, leaving (1 + r)ⁿ = 2, so doubling time does not depend on the amount invested. Taking logarithms of both sides gives n × ln(1 + r) = ln 2, so n = ln 2 ÷ ln(1 + r). Any base of logarithm works as long as it is the same on top and bottom, so log₁₀ 2 ÷ log₁₀(1 + r) gives the same answer. The result is usually not a whole number, because the doubling point falls partway through a year.
For small r, ln(1 + r) is close to r, so n ≈ 0.693 ÷ r, or 69.3 ÷ R with R in percent. That is the exact rule for continuous compounding. For yearly compounding, ln(1 + r) is a bit less than r, so the time is slightly longer, and 72 compensates well for rates around 8%. It also has many divisors, which makes mental arithmetic easy. At 7.1%, the exact time is about 10.1 years and 72 ÷ 7.1 is also about 10.1. At 24%, the rule gives 3 years against an exact 3.22.
Because each doubling takes the same time at a constant rate, growth over long periods can be counted in doublings. At 10%, money doubles about every 7.3 years, so over about 29 years it doubles four times and becomes sixteen times larger. At 7% it doubles about every 10.2 years, so over the same 29 years it grows less than eight times. A 3-point difference in return cuts the final outcome by more than half. This is the most practical lesson doubling time teaches about fees, taxes and choice of investment.
Usha's post office time deposit pays about 7.1% a year, and she wants to know exactly how many years it takes a deposit to double at that rate.
Answer: Exact doubling time 10.105 years
Assuming doubling time depends on the amount invested; it depends only on the rate.
Using the Rule of 72 at high rates and treating the result as exact.
Entering a monthly or quarterly rate as if it were yearly.
Ignoring tax and fees, which lower the effective rate and lengthen doubling time.
Using it for a SIP, where new money keeps being added, instead of a lump sum.
Checking claims such as 'your money doubles in 6 years'.
Comparing the long-run effect of two interest rates or fund expense ratios.
Estimating how quickly prices double at a given inflation rate.
Teaching logarithms and exponential growth with a real-world example.
How does this differ from the Rule of 72?
The Rule of 72 is a fast mental guess. This gives the exact answer using logarithms — useful when you need accuracy more than a quick mental estimate.