# APPLICATIONS OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS

(Interest Rate Word Problems)

1. To solve an exponential or logarithmic word problems, convert the narrative to an equation and solve the equation.

Problem 6: If you invested \$1,000 in an account paying an annual percentage rate compounded quarterly , and you wanted to have \$2,500 in your account at the end of your investment time, what interest rate would you need if the investment time were 1 year, 10 years, 20 years, 100 years?

Answer: 1 year = 102.97%, 10 years = 9.27%, 20 years = 4.61%, and 100 years 0.92%

Solution and Explanations:
Use the formula

where \$2,500 is the balance at the end of a certain time period, \$1,000 is the beginning investment, t is the number of years, and r is the annual percentage rate. The annual rate of r% is converted to a quarterly interest rate since the compounding is quarterly (4 times per year). Take the annual interest rate of 4% and divide by 4 to obtain the quarterly interest rate. The exponent is 4t because there are 4 compounding periods in every year. Therefore, 4t represents the number of compounding periods during t years.

To find the balance at the end of 1 year:

Step 1: Substitute 1 for t in the equation

to derive r:

Step 2: Divide both sides of the above equation by \$1,000:

Step 3: Take the natural logarithm of both sides of the above equation:

Step 4: Simplify the right side of the above equation using the third rule of logarithms:

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

Step 6: Simplify the left side of the above equation:

Step 7: Convert the above equation to an exponential equation with base e and exponent 0.29072682968:

Step 8: Simplify the left side of the above equation:

Step 9: Subtract 1 from both sides of the above equation:

Step 10: Multiply both sides of the above equation by 4:

This means the interest rate would have to be 102.97% (rounded) for the year.

Check:

This is a close enough check. Remember it will not check perfectly before we rounded the interest rate.

To find the balance at the end of 10 year:

Step 1: Substitute 10 for t in the equation

to derive r:

Step 2: Divide both sides of the above equation by \$1,000:

Step 3: Take the natural logarithm of both sides of the above equation:

Step 4: Simplify the right side of the above equation using the third rule of logarithms:

Step 5: Divide both sides of the above equation by 40:

Step 6: Simplify the left side of the above equation:

Step 7: Convert the above equation to an exponential equation with base e and exponent 0.0229072682968:

Step 8: Simplify the left side of the above equation:

Step 9: Subtract 1 from both sides of the above equation:

Step 10: Multiply both sides of the above equation by 4:

This means the interest rate would have to be 9.27% (rounded) per year for 10 years.

Check:

This is a close enough check. Remember it will not check perfectly before we rounded the interest rate.

To find the balance at the end of 20 year:

Step 1: Substitute 20 for t in the equation

to derive r:

Step 2: Divide both sides of the above equation by \$1,000:

Step 3: Take the natural logarithm of both sides of the above equation:

Step 4: Simplify the right side of the above equation using the third rule of logarithms:

Step 5: Divide both sides of the above equation by 80:

Step 6: Simplify the left side of the above equation:

Step 7: Convert the above equation to an exponential equation with base e and exponent 0.0114536341484:

Step 8: Simplify the left side of the above equation:

Step 9: Subtract 1 from both sides of the above equation:

Step 10: Multiply both sides of the above equation by 4:

This means the interest rate would have to be 4.61% (rounded) per year for 20 years.

Check:

This is a close enough check. Remember it will not check perfectly before we rounded the interest rate.

To find the balance at the end of 100 year:

Step 1: Substitute 100 for t in the equation

to derive r:

Step 2: Divide both sides of the above equation by \$1,000:

Step 3: Take the natural logarithm of both sides of the above equation:

Step 4: Simplify the right side of the above equation using the third rule of logarithms:

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

Step 6: Simplify the left side of the above equation:

Step 7: Convert the above equation to an exponential equation with base e and exponent 0.0022907682968:

Step 8: Simplify the left side of the above equation:

Step 9: Subtract 1 from both sides of the above equation:

Step 10: Multiply both sides of the above equation by 4:

This means the interest rate would have to be 0.92% (rounded) per year for 100 years.

Check:

This is a close enough check. Remember it will not check perfectly before we rounded the interest rate.

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