The population starts at 56,000.
After 1 year, it's (56,000) x (1.02) .
After 2 years, it's (56,000) x (1.02) x (1.02)
After 3 years, it's (56,000) x (1.02) x (1.02) x (1.02)
.
.
After ' n ' years, it's (56,000) x (1.02)ⁿ .
We want to know what ' n ' is when the population reaches 80,000 .
At that time . . .
80,000 = (56,000) x (1.02)ⁿ
Divide each side by 56,000 :
(80/56) = (1.02)ⁿ
Sorry, but now I have to take the log of each side:
log(80) - log(56) = n · log(1.02)
Divide each side by log(1.02) , and you have an expression for ' n ' :
n = [ log(80) - log(56) ] / log(1.02)
n = (1.903 - 1.748) / (0.0086)
n = 18 years
You can check this.
Just stuff (56,000) · (1.02)¹⁸ into your calculator.
I did that, and I got 79,981.8 , which is really awfully close to 80,000 .
