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.
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,...
To find the number of years, n, required for an investment to double in value, we need to solve the equation (1+r)^n = 2 for n.
Let's start by taking the natural logarithm of both sides of the equation:
ln[(1+r)^n] = ln(2)
Since ln(a^b) = b * ln(a), we can rewrite the equation as:
n * ln(1+r) = ln(2)
Now, we can use the approximation ln(1+r) ≈ r for small r. Since we are looking for an approximation, it is valid to use this approximation.
So, we can substitute r for ln(1+r) in the equation:
n * r = ln(2)
Finally, we solve for n by dividing both sides of the equation by r:
n = ln(2) / r
Therefore, the number of years, n, required for an investment to double in value is given by the formula:
n = ln(2) / r