If the population is tripled in 30 years, what is the annual growth rate?

Let x% be the annual growth rate.
Let a be the original population.

The population in 30 years.
a (1 + x%)^30 = 3 a
(1 + x%)^30 = 3
log[(1 + x%)^30] = log(3)
30 log(1 + x%) = log(3)
log(1 + x%) = (1/30) log(3)
1 + x% = 10^[(1/30) log(3)]
x% = 10^[(1/30) log(3)] - 1
x% = 0.0373
x = 3.73

The annual growth rate = 3.73%

To get the percentage, use 100 * (3^(1/30) - 1)

3 = e^(30 k) ===> k = (1/30) ln 3 ≈ 3.66%

3 = 1(1 + x)^30

3 = (1 + x)^30

ln(3) = 30(ln(1+x))

ln(1 + x) = ln(3) / 30

(1 + x) = e^(ln(3) / 30)

1 + x = 1.0373

x = 0.0373, or about 3.73 percent.

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