Doubling time formula
t₂ = ln(2) / ln(1 + r) ≈ 0.693 / r. The 0.693 constant is ln(2), the natural logarithm of 2, so the exact formula never changes, only the rate r you plug in. The continuous variant simplifies to ln(2) / k, skipping the step of converting r into a discrete growth factor first.
Rule of 70 comparison
The table below lines up the exact formula against the Rule of 70 shortcut across a range of common rates, so you can see exactly where the mental-math version starts to drift from the true value.
| Rate % | Exact doubling time | Rule of 70 |
|---|---|---|
| 1% | 69.66 | 70 |
| 2% | 35 | 35 |
| 3% | 23.45 | 23.33 |
| 4% | 17.67 | 17.5 |
| 5% | 14.21 | 14 |
| 6% | 11.9 | 11.67 |
| 7% | 10.24 | 10 |
| 8% | 9.01 | 8.75 |
| 9% | 8.04 | 7.78 |
| 10% | 7.27 | 7 |
| 12% | 6.12 | 5.83 |
| 15% | 4.96 | 4.67 |
| 20% | 3.8 | 3.5 |
Why doubling time matters
Doubling time turns an abstract percent growth rate into a tangible horizon you can picture. A growth rate of 1% sounds negligible but doubles a quantity every 70 periods, quadruples it every 140, and multiplies it more than a thousandfold over 700 periods. Because of this, doubling time, not the raw percentage, is the standard heuristic in demography, economics and biology.
Real-world doubling examples
Doubling time shows up far beyond finance. Bacteria such as the ones modeled on the bacterial growth calculator can double their population every 20 minutes under ideal lab conditions, so microbiologists count generations rather than percentages. Computing hardware followed Moore's Law for decades: transistor density on a chip doubled roughly every two years, compounding into a million-fold increase over 40 years. An investment account earning a steady 8% annual return doubles in ln(2) / ln(1.08) ≈ 9.01 years, so a 30-year career spans more than three full doublings. Population growth tells the same story at a slower pace. A country growing at 0.5% a year, typical of many developed economies, takes about 139 years to double. One growing at 3%, six times the rate, doubles more than five times faster, in just 23.4 years.
Doubling time and half-life
Doubling time has a mirror image on the decay side: half-life, the time for a quantity to fall to half its starting value. A process decaying continuously at rate k has half-life t½ = ln(2) / k, the same 0.693 constant that governs doubling, just applied to shrinkage instead of growth. Radioactive decay, drug elimination and cooling curves use half-life instead of doubling time because the quantity is falling rather than rising, but the underlying logarithm never changes.
FAQ
What is doubling time?
Doubling time is the number of periods a growing quantity needs to double in size, given by t₂ = ln(2) / ln(1 + r), which simplifies to about 0.693 / r when r is small. At a 4% growth rate, t₂ = ln(2) / ln(1.04) = 17.67 periods, meaning the quantity is twice its starting size after roughly 17 and two-thirds periods.
What is the Rule of 70?
The Rule of 70 estimates doubling time by dividing 70 by the percent growth rate, no logarithms required. At a 7% growth rate the estimate is 70 / 7 = 10 years, very close to the exact value of 10.24 years computed from ln(2) / ln(1.07). The shortcut works because 70 approximates 100 × ln(2) = 69.31, rounded up for easier mental division.
When does the Rule of 70 break down?
The Rule of 70 stays within about 2% of the exact answer for rates up to roughly 10%, then drifts further as compounding effects grow. At 20% growth the exact doubling time is 3.80 years while 70 / 20 gives 3.50 years, an error of nearly 8%. Above 15%, the Rule of 72 or the exact logarithm formula gives a more reliable answer.
Is the Rule of 70 better than the Rule of 72?
Neither is strictly better; they trade off different things. Rule of 72 tends to be more accurate near 8% growth because 72 divides evenly by more common rates, while Rule of 70 is mathematically closer to the true constant 100 × ln(2) = 69.31. For a 6% rate, Rule of 72 gives 12.0 years versus the exact 11.90, and Rule of 70 gives 11.67.
Can doubling time be applied to populations?
Yes, doubling time works for any quantity growing at a roughly constant percentage rate, including population. A country growing steadily at 2% a year doubles in about 35 years, since ln(2) / ln(1.02) ≈ 34.99. Global population growth, currently near 0.9% a year, implies a doubling time of about 77 years if that rate held steady, though real growth rates shift over time.
What is the doubling time for compound interest?
Compound interest follows the same t₂ = ln(2) / ln(1 + r) formula as any exponential growth. Money compounding annually at 6% doubles in 11.90 years, at 10% in 7.27 years, and at 12% in 6.12 years. Doubling the interest rate does not halve the doubling time because the relationship is logarithmic, not linear, unlike the straight-line Rule of 70 estimate.