Question

The number of years n required for an investment at interest rate r to double in value must satisfy (1+r)^n = 2. Using the approximation ln(1+r)=r valid for small r, find the number years n as a function of r.

Answer 1

take the ln of both sides
n ln(1+r)=ln2
n r = ln2
n= 1/r (.692)

so if r say = .02 (two percent interest)
n= 50*.692=35 years

If r= .10
n= 6.92 and the small r assumption starts to fail.