Population Doubling Time Formula
Calculate how long it takes a population to double using t_d = ln(2)/r.
Learn the doubling time formula with worked examples.
The Formula
The population doubling time formula calculates how long it takes for a population to double in size, given a constant growth rate. It applies to bacteria, animal populations, human demographics, and even cell cultures in the lab.
The formula is derived from the exponential growth equation N(t) = Nβ Γ e^(rt). Setting N(t) = 2Nβ and solving for t gives the doubling time.
This is sometimes called the "Rule of 70" in demography. If the growth rate is expressed as a percentage, doubling time β 70 / (growth rate in %). For example, a population growing at 2% per year doubles in about 35 years.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| t_d | Doubling time | Same unit as the growth rate's time base (hours, days, years) |
| ln(2) | Natural logarithm of 2, approximately 0.693 | β |
| r | Growth rate (as a decimal, not a percentage) | per time unit |
Example 1
E. coli bacteria divide every 20 minutes under optimal conditions. What is the growth rate, and does the formula confirm the 20-minute doubling time?
If the doubling time is 20 minutes, find r: r = ln(2) / t_d = 0.693 / 20
r = 0.0347 per minute
Verify: t_d = 0.693 / 0.0347 = 20 minutes β
The growth rate is 0.0347 per minute, confirming the 20-minute doubling time
Example 2
A country's population grows at 1.4% per year. How long until the population doubles?
Convert the percentage to a decimal: r = 1.4 / 100 = 0.014 per year
Apply the formula: t_d = 0.693 / 0.014
t_d β 49.5 years β the population will double in about 50 years
Example 3
A cell culture grows at a rate of 0.05 per hour. How long until it doubles?
Apply directly: t_d = 0.693 / 0.05
t_d = 13.86 hours β the culture doubles roughly every 14 hours
When to Use It
- Predicting bacterial colony growth in microbiology
- Demographic forecasting for human populations
- Cell culture planning in biomedical research
- Ecology β estimating wildlife population recovery timelines
- Public health β modeling disease spread during outbreaks