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.

Problem 2: Suppose you wanted to take out a mortgage for $100,000 with monthly payments at 9%, but you could only afford $800 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: 371.063 months or 30.92192 years or 30 years and 11 months. You would have 370 payments of $800 and the last payment would be $850.40. The interest paid over the term of the mortgage would be $216,850.40.

Solution and Explanations:

Step 1:

In the equation

substitute $100,000 for M (the mortgage amount), 12 for n (the number of payments per year, $800 for P (the monthly payment), and 9% for r (the annual interest rate)). You are solving for t (the term of the mortgage in years) or 12t( the term of the mortgage in months)

Step 2:

Multiply both sides of the above equation by 12:

Step 3:

Divide both sides of the equation by (.09)($100,000):

Step 4:

Multiply both sides of the equation by

to get

Step 5:

Divide both sides of the equation by 1.06666666667:

Step 6:

Subtract 1 from both sides of the above equation:

Step 7:

Multiply both sides by - 1:

Step 8:

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

Step 9:

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

Step 10:

Simplify the left side of the above equation:

Step 11:

Divide both sides of the above equation by: -0.0896641780644:

or 371.063064259 or 371 months.

If you would like to work another problem and review the answer and the solution, click on problem.

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