# 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 . At the end of the year you will have . This can also be written .
• At the end of the second year, you will have the you had at the beginning of the year plus the 12% interest on the . At the end of the second year you will have

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

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

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

• 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 which can be simplified to

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 .
• 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 . At the end of the month you will have . This can also be written .
• At the end of the second month, you will have the you had at the beginning of the month plus the 1% interest on the . At the end of the second month you will have

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

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

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

• 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

which can be simplified to

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 .
• 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 . At the end of the week you will have

This can also be written .

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

This can also be written

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

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

This can also be written

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

• 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 . 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 which can be simplified to

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:

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

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:

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

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

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

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

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 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.

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?

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?

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?

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?

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?

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?

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