Doubling Time and Half-Life

Doubling time is the time it takes a growing quantity to double, and half-life is the time it takes a decaying quantity to fall to half of its value. In brief: if a quantity doubles every T units of time, then after t units it is N₀ · 2^t/T. If it halves every T units, it is N₀ · (1/2)^t/T. A substance with a half-life of 5 days keeps 1/8 of its amount after 15 days.

These two numbers describe exponential processes in a way that is easy to understand: you do not need to know the growth factor, only the time to double or halve. The same formulas apply to bacteria, to money and to radioactive material. This guide explains the formulas, how to compute a doubling time from a growth rate with the rule of 70 and with logarithms, and gives a table of half-lives and remaining amounts.

What are the formulas for doubling time and half-life?

For a quantity with doubling time T_d and initial amount N₀, the amount after time t is N(t) = N₀ · 2^t/T_d. For a quantity with half-life T_h, it is N(t) = N₀ · (1/2)^t/T_h. The exponent t/T counts how many doubling times or half-lives have passed.

The two formulas are exponential functions with the base 2 or 1/2. Both can be converted to the usual form y₀ · b^t, with b = 2^1/T_d or b = (1/2)^1/T_h.

How do you calculate with a half-life?

Example 1: remaining amount

A sample of 80 g of a substance has a half-life of 6 hours. After 18 hours, t/T = 3, so the amount is 80 · (1/2)³ = 10 g. After 9 hours, t/T = 1.5 and the amount is 80 · (1/2)^1.5 = 80 · 0.3536 ≈ 28.3 g.

Example 2: time to reach a given amount

How long until 80 g falls to 5 g? The fraction is 5/80 = 1/16 = (1/2)⁴, so 4 half-lives, which is 24 hours. For a fraction that is not a power of 2, use logarithms: to reach 20 g, (1/2)^t/6 = 1/4, so t = 12 hours. To reach 30 g: (1/2)^t/6 = 0.375, so t/6 = ln 0.375/ln 0.5 = 1.415 and t ≈ 8.5 hours.

Example 3: finding the half-life from data

A substance decreases from 200 mg to 50 mg in 10 days. The fraction is 1/4, i.e., two half-lives in 10 days, so the half-life is 5 days.

How do you calculate a doubling time from a growth rate?

If a quantity grows by a factor b per time unit, it doubles when b^T = 2. Therefore T_d = ln 2/ln b = log_b 2. For a decay factor b < 1, the half-life is T_h = ln(1/2)/ln b = −ln 2/ln b.

Example 4: interest

Money grows by 5% per year, b = 1.05. T_d = ln 2/ln 1.05 = 0.6931/0.04879 ≈ 14.2 years.

Example 5: population

A town grows by 2% per year. T_d = ln 2/ln 1.02 = 0.6931/0.01980 ≈ 35.0 years.

Example 6: decay

A medicine decays by 12% per hour, b = 0.88. T_h = −ln 2/ln 0.88 = −0.6931/(−0.12783) ≈ 5.42 hours.

What is the rule of 70?

For small growth rates, the doubling time in periods is approximately 70 divided by the percentage rate: T_d ≈ 70/r. The number 70 comes from 100 · ln 2 ≈ 69.3, rounded to a number with more divisors. A rate of 5% gives about 14 periods, which matches the exact 14.2. At 2%, the estimate is 35, matching 35.0. At 10%, the rule gives 7 and the exact value is ln 2/ln 1.1 ≈ 7.27. The rule is less accurate for large rates: at 50% it gives 1.4, but the exact value is 1.71.

Some texts use the rule of 72, since 72 has many divisors (2, 3, 4, 6, 8, 9, 12) and works well for rates around 8%. Both are approximations, not exact laws.

Remaining fraction after n half-lives

Where do these ideas come from?

The concept of half-life was developed in the study of radioactivity at the beginning of the twentieth century. Ernest Rutherford and Frederick Soddy described the exponential decay of radioactive substances around 1902 and 1903, and the term half-life was introduced soon afterwards. Doubling time is a much older idea, related to compound interest problems found on Babylonian tablets, which asked how long it takes for a sum to double.

Where are they used today?

Carbon-14 has a half-life of about 5,730 years, which allows archaeologists to date organic material up to roughly fifty thousand years old. Doctors use the half-life of a drug in the body to determine the dosing interval. Financial planners use the rule of 72 to estimate how long it takes for savings to double, and public health specialists use the doubling time of cases to describe the speed of an epidemic.

Which mistakes are common?

Assuming that two half-lives remove everything

After two half-lives, 25% remains. The amount never reaches exactly zero in the model.

Mixing up time and the number of periods

The exponent is t/T, the number of half-lives, not t.

Using the rule of 70 for high rates

The rule is an approximation that works for rates below about 10%.

Applying a constant doubling time when growth is limited

Real systems stop growing exponentially when resources run out.