APPLICATIONS OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS

(Amortization Word Problems)

To solve an exponential or logarithmic word problem, convert the narrative to an equation and solve the equation.
There is a relationship between the mortgage amount, the number of payments, the amount of the payment, how often the payment is made, and the interest rate. The following formulas illustrate the relationship:

where P = the payment, r = the annual rate, M = the mortgage amount, t = the number of years, and n = the number of payments per year.

Example 2: Suppose you wanted to take out a mortgage for $75,000 with monthly payments at 7%, but you could only afford $450 per month payments. How long would you have to make payments to pay off the mortgage, and how much interest would you pay for this payment period?

Answer: It would take you 616.10755485 months to pay off the mortgage, roughly 51 years and 5 months. The bank would either have you pay 616 payments of $450 per month and one last payment of $48.40 or your 616th payment would be $498.40.

Solution and Explanations:

Step 1: In the equation

substitute $75,000 for M (the mortgage amount), 7% for r (the annual interest rate), 12 for n (the number of payments per year, and $450 for P (the mortgage payment). You are solving for t (the number of months that you must make payments)

Step 2: Multiply both sides of the above equation by 12:

Step 3: Multiply both sides of the above equation by

to get:

Step 4: Divide both sides of the above equation by 5,400:

Step 5: Subtract 1 from both sides of the above equation:

Step 6: Multiply both sides of the equation by - 1:

Step 7: Take the natural log of both sides of the above equation:

Step 8: Simplify the left side of the above equation using the third rule of logarithms:

Step 9: Divide both sides by Ln(1.00583333333) and simplify:

Step 10: Divide both sides by - 1:

Step 11: The 12t stands for the number of months in t years or 616 months plus part of a month. The number of years is 51.3422962552 or a little more than 51 years to pay off the mortgage.
The amount that you will pay over this term is (616)($450) = $277,200 +1(.107555063)(450) or $48.39977835 rounded to $48.40. Normally, the bank will add this extra amount on the 616th payment. Sometimes they will have a 617th pay of $48.40. The total amount paid over the mortgage period is $277,248.40.
Everything over the $75,000 mortgage is interest. The interest you will pay over the 51 plus years is $277,248.40 - $75,000 = $202,248.40

If you would like to review another example, click on Example.

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