# 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.

Example 2: A \$5,000 investment is made in a trust fund at an annual percentage rate of 10%, compounded annually. Predict the balance in the account after 5 years. How long will it take the investment to reach \$15,500? Suppose that another bank promised you that your account would reach \$15,500 in 10 years, what annual interest would the second bank be paying?

Explanation and Solution:

• Balance at the end of the first year: The trust account began the year with the \$5,000 investment and will have a balance equal to the \$5,000 plus the 10% interest on the \$5,000 at the end of the first year.

or .

• Balance at the end of the second year: The trust account began the year with a balance of \$5,500 investment and will have a balance equal to the \$5,500 plus the 10% interest on the \$5,500 at the end of the second year.

or .

• Note that in terms of the initial investment, the above steps could be written

• Balance at the end of the third year: The trust account began the year with a balance of \$6,050 investment and will have a balance equal to the \$6,050 plus the 10% interest on the \$6,050 at the end of the third year

or .

• Note that in terms of the initial investment, the above steps could be written

• Balance at the end of the fourth year: The trust account began the year with a balance of \$6,655 investment will have a balance equal to the \$6,655 plus the 10% interest on the \$6,655 at the end of the second year

or

• Note that in terms of the initial investment, the above steps could be written

• Balance at the end of the fifth year: The trust account began the year with a balance of \$7,320.50 investment will have a balance equal to the \$7,320.50 plus the 10% interest on the \$7,320.50 at the end of the fifth year

or

• Note that in terms of the initial investment, the above steps could be written

• If you deposit \$5,000 in an account that pays 10% per year with annually compounding, and you left the money in the account for 5 years, you would have a balance of \$8,052.55 at the end of the 5 years.
• The second part of the problem was to estimate how many years it would take for the account to teach \$15,500.
• In this problem, you know the starting amount, the ending amount, and the interest rate, what you are trying to determine is the time

• Divide both sides by \$5,000:

• Take the Log of both sides:

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

• Divide both sides of the above equation by Log(1.10):

• This means that it takes almost 12 years for your account to reach \$15,500.
• The third part of the problem was to estimate the interest rate if your \$5,000 reached \$15,500 in 10 years.

• In this problem, you know the starting amount, the ending amount, and the time, what you are trying to determine is the annual interest rate

• Divide both sides of the above equation by \$5,000:

• Take the Log of both sides of the above equation:

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

• Divide both sides of the above equation by 10:

• Calculate the left side of the above equation:

• Rewrite the logarithmic equation as an exponential equation with base 10 and exponent 0.04913616938834:

• Subtract 1 from both sides of the above equation:

• Simplify the left side of the above equation:

rounded to 0.1198

• This means that for the \$1,000 to reach \$15,500 in 10 years, the bank would have to pay 11.98% interest, compounded annually.

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

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