APPLICATIONS OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS

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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