Question

there are 50 rabbits on a farm. They are increasing at the rate of 12% every month. How many months will it be until there are 1000 rabbits?

Answer 1

each month there will be 112% of the previous month so the multiplier is 1.12

basically it's trial and error

after 1 month there will be 50 x 1.12 = 56

after 2 months there will be 50 x 1.12^2 = 63

after 20 months there will be 50 x 1.12^20 = 482

after 25 months there will be 50 x 1.12^25 = 850

after 30 months there will be 50 x 1.12^30 = 1498

this is too many so try lower values

after 28 months there will be 50 x 1.12^28 = 1194

after 27 months there will be 50 x 1.12^27 = 1066

after 26 months there will be 50 x 1.12^26 = 952

Answer 2

Trial and error aren't necessary.

Increasing 12% per month means that at any time,
there are (1.12) x (the number there were a month earlier).

After, say, 'M' months, the number has grown by (1.12)^M power.

We want to know what 'M' has to be so that 50(1.12)^M = 1,000

Divide each side by 50 : (1.12)^M = 20

Take the log of each side: M log(1.12) = log(20)

Divide each side by log(1.12): M = log(20) / log(1.12)

M = 1.301 / 0.0492 = 26.44 months