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 1: A $1,000 deposit is made at a bank that pays 12% compounded annually. How much will you have in your account at the end of 10 years?

Explanation and Solution:

  • At the end of the first year, you will have the $1,000 you had at the beginning of the year plus the interest on the $1,000 or tex2html_wrap_inline162 . At the end of the year you will have tex2html_wrap_inline164 . This can also be written tex2html_wrap_inline166 .
  • At the end of the second year, you will have the tex2html_wrap_inline166 you had at the beginning of the year plus the 12% interest on the tex2html_wrap_inline166 . At the end of the second year you will have

    displaymath172

    This can also be written tex2html_wrap_inline174 . Another way of writing this is to write the balance at the end of the second year as tex2html_wrap_inline176 .

  • At the end of the third year, you will have the tex2html_wrap_inline176 you had at the beginning of the year plus the 12% interest on the tex2html_wrap_inline176 . At the end of the third year you will have

    displaymath182

    This can also be written tex2html_wrap_inline184 . Another way of writing this is to write the balance at the end of the third year as tex2html_wrap_inline186 .

  • By now you should notice some common things in each end-of-year balance. For one thing, the exponent is the same as the year. The base is always 1 + rate or 1 + .12. The $1,000 will always stay the same in the formula.
  • Now we can write the balance at the end of 10 years as tex2html_wrap_inline188 which can be simplified to

    displaymath190

    rounded to $3,105.85.

  • Example 2: An $1,000 deposit is made at a bank that pays 12% compounded monthly. How much will you have in your account at the end of 10 years?

    Explanation and Solution:

  • In this example the compounded is monthly, so the interest rate has to be converted to a monthly interest rate of tex2html_wrap_inline192 .
  • At the end of the first month, you will have the $1,000 you had at the beginning of the month plus the interest on the $1,000 or tex2html_wrap_inline194 . At the end of the month you will have tex2html_wrap_inline196 . This can also be written tex2html_wrap_inline198 .
  • At the end of the second month, you will have the tex2html_wrap_inline198 you had at the beginning of the month plus the 1% interest on the tex2html_wrap_inline198 . At the end of the second month you will have

    displaymath204

    This can also be written tex2html_wrap_inline206 . Another way of writing this is to write the balance at the end of the second month as tex2html_wrap_inline208 .

  • At the end of the third month, you will have the tex2html_wrap_inline208 you had at the beginning of the month plus the 1% interest on the tex2html_wrap_inline208 . At the end of the third month you will have

    displaymath214

    This can also be written tex2html_wrap_inline216 . Another way of writing this is to write the balance at the end of the third month as tex2html_wrap_inline218 .

  • By now you should notice some common things in each end-of-month balance. For one thing, the exponent is the same as the number of months that have passed. The base is always 1 + rate or 1 + .01. The $1,000 will always stay the same in the formula.
  • There are 120 month in 10 years; therefore, we write the balance at the end of 10 years as

    displaymath220

    which can be simplified to

    displaymath222

    rounded to $3,300.39.

  • Example 3: An $1,000 deposit is made at a bank that pays 12% compounded weekly. How much will you have in your account at the end of 10 years?

    Explanation and Solution:

  • In this example the compounded is weekly, so the interest rate has to be converted to a weekly interest rate of tex2html_wrap_inline224 .
  • At the end of the first week, you will have the $1,000 you had at the beginning of the week plus the interest on the $1,000 or tex2html_wrap_inline226 . At the end of the week you will have

    displaymath228

    This can also be written tex2html_wrap_inline230 .

  • At the end of the second week, you will have the tex2html_wrap_inline230 you had at the beginning of the week plus the tex2html_wrap_inline224 interest on the tex2html_wrap_inline230 . At the end of the second week you will have

    displaymath238

    This can also be written

    displaymath240

    Another way of writing this is to write the balance at the end of the second week as

    displaymath242

  • At the end of the third week, you will have the tex2html_wrap_inline244 you had at the beginning of the week plus the tex2html_wrap_inline224 interest on the tex2html_wrap_inline244 . At the end of the third week you will have

    displaymath250

    This can also be written

    displaymath252

    Other way of writing this is to write the balance at the end of the third week as tex2html_wrap_inline254 .

  • By now you should notice some common things in each end-of-week balance. For one thing, the exponent is the same as the week. The base is always 1 + rate or tex2html_wrap_inline256 . The $1,000 will always stay the same in the formula.
  • There are 520 weeks in 10 years; therefore, we write the balance at the end of 10 years as tex2html_wrap_inline258 which can be simplified to

    displaymath260

    rounded to $3,315.53.

  • Example 4: An $1,000 investment is made in a trust fund at an annual percentage rate of 12%, compounded monthly. How long will it take the investment to reach $2,000?

    Answer: It would take about 5 years and 10 months for the investment to reach $2,000.

    Explanation and Solution:

    Step 1: The annual percentage rate is the rate that you would receive if the interest was calculated at the end of the year. This means there was no compounding during the year.
    Step 2: Determine what the interest rate would be per month by dividing the 12% by 12 months:

    displaymath262

    Step 3: From Example 2 above, we know that we can find the balance after t years as follows:

    displaymath264

    We use 12 t because there are 12 months in every year.

    Step 4: Replace the right side of the above equation with $2,000:

    displaymath268

    Step 5: We must isolate the exponential term; therefore, divide both sides by $1,000:

    displaymath270

    Step 6: Take the natural logarithm of both sides of the above equation:

    displaymath272

    Step 7: Simplify the left side of the above equation:

    displaymath274

    Step 8: Divide both sides of the above equation by tex2html_wrap_inline276 :

    displaymath278

    Step 9: Note that 5.80505974113 years can be written 5 Years + 0.80505974113 Years. If you multiply 0.80505974113 years by 1 in the form tex2html_wrap_inline280 you get 9.6607 months. This indicates that it takes 5 years and about 10 months for the $1,000 to reach $2,000.

    If you would like to see more examples on interest rates, click on Example.

    Work the following problems. If you want to check your answer and solution click on Answer.

    1.
    If you invested $1,000 in an account paying an annual percentage rate (quoted rate) of 12%, compounded quarterly, how much would you have in you account at the end of 1 year, 10 years, 20 years, 100 years?

    Answer.

    2.
    If you invested $1,000 in an account paying an annual percentage rate (quoted rate) of 12%, compounded weekly, how much would you have in you account at the end of 1 year, 10 years, 20 years, 100 years?

    Answer.

    3.
    If you invested $1,000 in an account paying an annual percentage rate (quoted rate) compounded daily (based on a bank year of 360 days) 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.

    4.
    If you invested $1,000 in an account paying an annual percentage rate (quoted rate) of 12%, compounded hourly (based on a bank year of 360 days), how much would you have in you account at the end of 1 year, 10 years, 20 years, 100 years?

    Answer.

    5.
    If you invested $1,000 in an account paying an annual percentage rate (quoted rate) of 12%, compounded continuously, how much would you have in you account at the end of 1 year, 10 years, 20 years, 100 years?

    Answer.

    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.

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