Often times when you read something like “*Annual increases at a 10% rate would lead to the doubling of prices every seven years*“, you might be wondering how one would go about calculating the length of time required for a single cash flow(present value) to reach a certain amount(future value) based on the given rate.

We all know that a future value can be calculated using the compounding interest formula as shown below:

with:

- is the present value
- is the future value
- r is the nominal annual interest rate
- n is the number of years

Now, suppose you know the given rate , the present value and the future value . How can you calculate ?

From the above formula we can derive that:

Taking the logarithm on both sides we get:

Simplifying the exponent on the right side of the above equation, we have:

Finally, solving the above equation for , we get:

So if we want to know how long it takes for a value to double given a rate of a year, we would simply do: