Doubling Time
question
Israel has currently (2021) a population of 9.2 million, and a growth rate of 1.8% per year. How long will it take for the population to double, assuming a fixed growth rate?
Suppose you have a process that can be described by exponential growth. It could be anything: interests on an investment, the early phases of infection in a pandemic, whatever.
It is often convenient to have an idea how fast is the growth by answering the question:
How long will it take for
to double in size, given a growth of % per year?
The rule of thumb I learned a while back is the following:
Doubling time =
(in years)
Of course, the time unit could be anything you like, I’ll deal here with years for simplicity’s sake.
The answer to the question about Israel in the box above is about 39 years (70 divided by 1.8), but why?!
Let’s call
Assume that after
Cancelling
We now take the natural logarithm of both sides in Equation 3, yielding:
Note that we took
That surely explains the number 70 in the rule of thumb! Because of the properties of logarithms, we put the number 100 as the exponent of the parenthesis:
We are very close to the end! We now remind ourselves that we learned in Calculus the definition of the exponential function:
Because the number 100 is “quite big”, we will approximate the parenthesis inside the logarithm in Equation 6 with the exponential function in Equation 7, yielding
The logarithm is the inverse function of the exponential, therefore
Finally, solving for
We have thus shown that the number of years
The exact number, without any approximations, would be
conclusion
👍 Very impressive rule of thumb 👍