Doubling time formula
The exact doubling time for compound growth is t = ln(2) ÷ ln(1 + r), where r is the growth rate per period as a decimal. A faster mental shortcut is the rule of 70: divide 70 by the percentage growth rate to approximate the doubling time.
The formula follows from setting the growth equation equal to twice the starting amount: (1 + r) raised to the power t equals 2. Taking natural logarithms of both sides turns the exponent into a multiplier, and dividing gives t on its own. Notice that the starting size cancels out entirely — doubling time depends only on the rate. For bacteria and cell cultures specifically, the cell doubling calculator uses the same relationship.
How to use this calculator
Enter the growth rate as a percentage per period. The tool returns the exact doubling time and the rule-of-70 estimate side by side. To then project the actual future size, use the population growth calculator.
Whatever period your rate refers to is the period the answer comes back in: a rate per year gives years, a rate per hour gives hours.
Worked example
At 7% growth: t = ln(2) ÷ ln(1.07) = 0.6931 ÷ 0.0677 ≈ 10.24 years. The rule of 70 gives 70 ÷ 7 = 10 years — a close approximation.
The gap between the two answers widens as the rate climbs. At small rates the shortcut is accurate to within a fraction of a period, so use the exact logarithmic result whenever the rate is large. The formula rests on natural logs, which the logarithm calculator evaluates for any base.
Frequently asked questions
- What is doubling time?
- The time it takes for a quantity growing at a constant rate to become twice its starting size.
- What is the rule of 70?
- A shortcut for doubling time: divide 70 by the percentage growth rate. It approximates the exact logarithmic formula and works well for small rates.
- Why is the exact answer different from the rule of 70?
- The rule of 70 is an approximation. The exact formula uses natural logarithms; the two agree closely at low growth rates and diverge slightly at high rates.
- Can I use this for cells or money?
- Yes. Doubling time applies to any quantity with a constant percentage growth rate, including cell cultures, populations, and compound interest.
- What happens if the growth rate is zero or negative?
- There is no doubling time. At zero growth the quantity never doubles, and at a negative rate it shrinks instead — the matching concept there is half-life, the time taken to fall to half the starting value.