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

ln (1+r)^n = n * ln(1+r)
for small r, ln(1+r) =~ r, so
n*ln(1+r) =~ n*r

if n*r = ln 2,
n = ln2/r

for some small r, starting at 1%, that is about
69,35,23,17,14,12,10,9,8,...