Doubling Time Formula
Calculate how long it takes a population or quantity to double using the doubling time formula t = ln(2)/r.
With worked examples.
The Formula
The doubling time tells you how long it takes for something growing exponentially to double in size. It applies to bacteria dividing, human populations growing, investments compounding, or any quantity with a constant growth rate.
The formula is derived from the exponential growth equation N(t) = N₀ × ert. Setting N(t) = 2N₀ and solving for t gives the doubling time.
A useful approximation is the Rule of 70: if the growth rate is given as a percentage, the doubling time is approximately 70 divided by the growth rate percentage. For example, 7% growth doubles in about 10 years.
Calculator
Variables
| Symbol | Meaning |
|---|---|
| td | Doubling time (in the same time units as r) |
| ln(2) | Natural logarithm of 2 ≈ 0.6931 |
| r | Growth rate (as a decimal, e.g., 0.05 for 5% growth) |
Related: Population Growth Equation
Once you know the doubling time, you can predict the population at any future time t. After one doubling time, the population is 2×. After two doubling times, it is 4×. After three, it is 8×, and so on.
Example 1
E. coli bacteria divide every 20 minutes under ideal conditions. Starting with 1,000 bacteria, how many will there be after 3 hours?
Doubling time: td = 20 minutes
Number of doublings in 3 hours: 180 / 20 = 9 doublings
N = 1,000 × 2⁹ = 1,000 × 512
N = 512,000 bacteria after 3 hours
Example 2
A country's population grows at 1.2% per year. How long will it take the population to double?
Convert rate to decimal: r = 0.012
Apply the formula: td = ln(2) / r = 0.6931 / 0.012
td ≈ 57.8 years (or using the Rule of 70: 70 / 1.2 ≈ 58.3 years — very close)
Example 3
A bacterial culture doubles every 45 minutes. What is the growth rate per hour?
Convert doubling time to hours: td = 45/60 = 0.75 hours
Rearrange: r = ln(2) / td = 0.6931 / 0.75
r ≈ 0.924 per hour (92.4% growth per hour)
When to Use It
Use the doubling time formula for any quantity growing exponentially.
- Predicting bacterial population growth in microbiology
- Estimating how quickly a country's population will double
- Calculating how long an investment takes to double (Rule of 72 in finance)
- Modeling the spread of viral content or diseases in early stages
- Understanding the pace of technological growth (Moore's Law)