How do you figure this on a scientific calculator?

Question

A population grows from 37,172,015 to 50,000,000 in 14 years, what keys do you need to punch in to find the per annum growth rate? I tried doing (50,000,000 - 37,172,015) ^ (1/14) but that just gave me a big number.

Answer 1

The formula for population growth is P(n) = P(0)*(1+r)^n, where r is the fractional growth per year, n=no of years, P(0) is the starting value. To solve for r, take log of both sides to get

log[P(n)] = log[P(0)] +n*log(1+r)

Then log(1+r) = {log[P(n)] - log[P(0)]}/n = log[P(n)/P(0)]/n

Raise both sides to the power of 10

1+r = 10^{log[P(n)/P(0)]/n}

r = 10^{log[P(n)/P(0)]/n} - 1

So enter P(n), divide by P(0), take log of the quotient. Divide this by n and take 10 to the power of this result, then subtract 1.