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.

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.

To find the number of years, n, required for an investment to double in value given an interest rate, r, we can use the equation (1+r)^n = 2.

Let's start by taking the natural logarithm of both sides of the equation to solve for n:

ln((1+r)^n) = ln(2)

Using the property of logarithms, we can rewrite the equation as:

n * ln(1+r) = ln(2)

Now, since we have an approximation that ln(1+r) ≈ r for small values of r, we can substitute this approximation into the equation:

n * r = ln(2)

Finally, to solve for n, we divide 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 as a function of the interest rate, r, is given by:

n = ln(2) / r