Activity 7.2, Estimations, Scientific Notation, and Properties of Logarithms

In this activity, you will use properties of logarithms to investigate and answer questions about an investment.

1. Let's investigate an equation that relates to compound interest. Remember the example in class about the best summer job? It was agreed that earning a $10,000 salary for being a lifeguard, with a 6% raise each summer, would be pretty awesome. So, the initial summer you would earn $10,000, and the next summer your salary would be
10,000 + 0.06*10,000 = 10,000 + 600 = $10,600.00. This is just 1.06*10,000.

What would your salary be in each of the next two summers?

2. We will set up an Excel worksheet to help calculate the salary, using the following instructions.

3. a. Open a new sheet in your Excel workbook and label column A as Summer Number and column B as Salary.

3. b. Enter numbers 0 through 20 in column A, starting in cell A2, using Excel's auto-fill (we did this previously).

Enter your initial salary, $10,000, in cell B2. Then to fill the rest of the column, you will enter enter a recursive formula. Start with a formula in B3 that gives the amount in B3 in terms of the amount already in B2 (look at the formula/hint in #1 above), and then drag to fill in the rest of the column.

Copy and paste this table into your MS Word document.

4. What formula did you enter in cell B3?

5. What amount did you obtain for the salary in the last row, or 20th summer after the first?

6. The salary after n years (still assuming an initial salary of $10,000 and a yearly 6% raise) can be given by a single formula in terms of n:

Salary after n years = 10000 * (1.06)ⁿ

This formula is an exponential function, since the variable n is in the exponent, and it differs from the recursive calculations you did in column B, where the new salary is simply calculated from the old salary. In column C of your spreadsheet, enter the values obtained using this formula. (To do this, place your cursor in cell C2 and enter the appropriate formula, =10000*(1.06)^A2. Then drag down to year 20). Paste this updated table into your MS Word document.

7. Using the previous formula, you can find the salary after any number of years, without using a spreadsheet. You can also calculate the number of years it would take for the salary to reach any given amount. For example, to find out how many years it would take for the account to reach $20,000, you need to find the value of n so that 20,000 = 10000 * (1.06)ⁿ .

To do this, solving for the n required to reach $20,000, we need to get n out of the exponent. If we apply the logarithm function to both sides of the equation, we get the following:

log( 20,000 ) = log( 10,000) + log( 1.06ⁿ ). Then what property of logarithms would we apply to get the following equation?

log( 20,000 ) = log( 10,000) + n*log( 1.06 )

8. We can re-write log(20,000) as log(2 * 10,000). What property of logarithms is applied for:
log(2 * 10,000) = log(2) + log(10000).

9. So the equation in #7 becomes:
log(2) + log(10,000) = log(10,000) + n*log(1.06).

Because log( 10,000 ) = 4 and log( 2 ) = 0.30, we have:
0.30 + 4 = 4 + n*log( 1.06 )

Use algebra to solve the above equation for n in terms of log(1.06) (that means to leave the term as log(1.06), do NOT evaluate it yet) , and show your steps in your MS Word document. Also, explain why log( 10,000 ) = 4.

10. Use a calculator or Excel to calculate an approximate value of log(1.06), and then use this to approximate n, which is the number of summers after the initial summer for the salary to reach $20,000.

n = _______________

11. Use the same approach that you used to above, in #7 - #10, to find the number of summers after the initial summer it would take for the salary to reach $100,000, and show all the steps in your MS Word document.

Summary
In this activity, you used Excel to calculate compound interest and used logarithms to solve equations to determine growth in an exponential function.