SOLUTION: How long will it take, to the nearest month, for $2500 to grow to $4000, if it is invested at 7%, compounded monthly?

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Question 282951: How long will it take, to the nearest month, for $2500 to grow to $4000, if it is invested at 7%, compounded monthly?
Answer by jsmallt9(3758)   (Show Source): You can put this solution on YOUR website!
The formula for compound interest is:

where
P = Principal (the amount of the initial investment)
r = annual rate of interest (as a decimal or fraction)
n = the number of compounding periods per year
t = the total number of compounding periods of the investment
A = Amount (the value of the investment after t compounding periods)

Since your rate is 7% and the investment is compounded monthly, your "r" is 0.07 and your n is 12:

The expression in the parentheses simplifies as follows:



You are asked to find how long it will take for an investment of 2500 to grow to 4000. So A = 4000 and P = 2500:

Now we solve for t. We'll start by isolating the base and its exponent. Divide both sides by 2500:




Solving for a variable in an exponent usually involves logarithms. So we'll find the logarithm of each side. (Any base of logarithm can be used. But if you want a decimal approximation of the answer it is best to use a base your calculator "knows" (like base 10 or base e (ln))). We'll use base 10:

Now we can use a property of logarithms, , to move the exponent out in front. (This property, with its ability to change an exponent into a coefficient, is the very reason we use logarithms on equations where the variable is in an exponent.)

Now we can divide both sides by :

This is an exact expression of the answer. You probably want a decimal approximation so use your calculator on this. If your calculator has keys for parentheses then you can pretty much type in what you see with the parentheses. If not, then
  1. Divide 12.07 by 12
  2. Find the log of the answer from step 1.
  3. Find the log of 1.6
  4. Divide the result of step 3 by the result of step 2

The answer you get will be the approximate number of compounding periods (which are months in this problem) it will take to reach $4000.
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