Question

What does the interest rate have to be, if a bank pays interest, compounded continuously and the amount doubles in 5 years?

Answer 1

Let the original amount be A.  Double that would be 2A.

Then, 2A = Ae^(r*5), or 2 = e^(5r).

Taking the natural log of both sides:  ln 2 = 5r.  Then r = (ln 2)/5 = 0.14

The interest rate would have to be 13.86% per year, written here as r=0.14.