How Do You Solve For Time in the Compound Interest Formula?

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.

This problem requires the use of the compound interest formula,

compound_01

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

compound_02_01

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(yx) = x logy. 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:

compound_02_02

Now take the logarithm of both sides of the equation:

compound_02_03

This gives us

compound_02_04

or n ≈ 9.21 years.

In WolframAlpha, we could evaluate the logs as follows.

wolfram_log