A population growing at a steady rate increases slowly at first, but in absolute numbers, it can grow explosively after a time. For example, a population of just 10 people, growing at a rate of 10 per cent per year, will double to 20 in about seven years. In 20 years, it grows almost eight times as large. You can use an exact formula to calculate a population's doubling time for a given growth rate. You can also use a rule of thumb to make a fast approximation. You'll need a scientific calculator to compute logarithms for the exact method.

- Skill level:
- Easy

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### Things you need

- Scientific calculator

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## Instructions

- 1
Determine the logarithm of 2. This is the numerator of a fraction you will evaluate below. You can use either a natural logarithm or base 10. If you use base 10, then enter "2" on your calculator, and press the "Log" key to determine that Log 2 = 0.301. Find the natural log by entering "2," and pressing the "Ln" key. The answer is 0.693.

- 2
Define the annual growth percentage of the population as P. Divide P by 100 to convert it from a percentage to a decimal. For example, if P = 10, then P/100 = 0.1.

- 3
Add 1 to the number computed in Step 2. In the example, 1 = + 0.1 = 1.1.

- 4
Find the logarithm of the number you computed in Step 3. In the example, enter "1.1" and press the "Log" key to find an answer of 0.0414. If you're using natural logs, press the "Ln" key instead to obtain an answer of 0.0953.

- 5
Divide log 2, which you determined in Step 1, by the logarithm you found in Step 4. In the example, 0.301/0.0414 = 7.27. The doubling time for a population with a 10 per cent annual growth rate is 7.27 years. If you use natural logs, your answer is 0.693/0.0953 = 7.27.

#### Tips and warnings

- You can obtain a fairly accurate approximation of the doubling time by applying the "rule of 72." Divide 72 by the percentage growth rate. In the example, the growth rate is 10 per cent. Divide 72 by 10 to obtain an answer of 7.2, which compares nicely to the exact answer of 7.27.
- Due to the nature of logarithms, you can use any base in your calculations because you're dividing one logarithm by another. That's why you got the same answer using natural logs as with base 10.
- You determined a doubling time in years, but your calculations hold for any units of time as long as you use the growth rate for that unit of time. For example, if the growth rate in your example is ten per cent per week, then the doubling time is 7.27 weeks instead of 7.27 years.