Question

The population of a country doubled in 72 years. What is the exponential growth rate?

Answer 1

y = a(1+r)^x is the exponential growth function.
a is the initial amount, x is the time, r is the growth rate, y is the amount at time x.
put x = 0
y = a(1+r)^0 = a
So the initial population is a.
The population doubled in 72 years.
That is, when x = 72, y = 2a
2a = a(1 + r)^72
divide both sides by a
2 = (1+r)^72
take log on both sides:
log(2) = 72log(1+r)
log(1+r) = log(2)/72 = 0.004181
1 + r = 10^0.004181
r = 10^0.004181 - 1
r = 0.009674

Growth factor r = 0.0097 or 0.97% (can be rounded to 1%)

Answer 2

Yes. The "rule of 72" says that growth factor (%) (time period) = 72 for doubling, almost exactly. 1% doubles in 72 years; 2% in 36 years; 6% in 12 years, etc. Handy. Here it would have given a 1% growth factor, just as the detailed calculation showed.