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.

    displaymath50

    or tex2html_wrap_inline52 .

  • 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.

    displaymath54

    or tex2html_wrap_inline56 .

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

    displaymath58

  • 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

    displaymath60

    or tex2html_wrap_inline62 .

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

    displaymath64

  • 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

    displaymath66

    or tex2html_wrap_inline68

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

    displaymath70

  • 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

    displaymath72

    or tex2html_wrap_inline74

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

    displaymath76

  • 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

    displaymath78

  • Divide both sides by $5,000:

    displaymath80

  • Take the Log of both sides:

    displaymath82

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

    displaymath84

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

    displaymath86

  • 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

    displaymath88

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

    displaymath90

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

    displaymath92

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

    displaymath94

  • Divide both sides of the above equation by 10:

    displaymath96

  • Calculate the left side of the above equation:

    displaymath98

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

    displaymath100

  • Subtract 1 from both sides of the above equation:

    displaymath102

  • Simplify the left side of the above equation:

    displaymath104

    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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    Author: Nancy Marcus

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