In another MathFAQ, I looked at how you can find the rate in the compound interest formula. Now let’s look at an example where we solve for the number of years *n*. This problem is different because what we are looking for appears in a power.

**Problem** Suppose $5000 is deposited in an account that earns 2% compound interest that is done annually. In how many years will there be $6000 in the account.

**Solution** This problem requires the use of the compound interest formula,

This formula applies when interest is earned on an annual basis and the interest is earned once a year.

Let’s look at the quantities in the problem statement:

- $5000 is deposited in an account >
*P*= 5000 - that earns 2% compound interest that is done annually >
*r*= 0.02 - Will there be $6000 in the account >
*A*= 6000

Putting these values into the formula above gives us

Unlike other problems where we solve for P or r, here we need to solve for the power in the right hand side, n. Solving for a value in the power requires the property of logarithms, log(*y ^{x}*) =

*x*log

*y*. It allows us to move the

*n*in the power and change it to a multiplier. But before we can apply this property, we isolate the factor containing the

*n*:

Now take the logarithm of both sides of the equation:

This gives us

or *n* ≈ 9.21 years.

In WolframAlpha, we could evaluate the logs as follows.