Training Camp

In Lesson 6A, you reviewed the laws of exponents, and you used them to simplify exponential expressions and solve exponential equations. Earlier in this unit, you learned that logarithms are exponents. Does this mean there are related laws for working with logarithms? In the following investigation, you will explore this possibility.

Complete the questions to investigate the laws of logarithms. Click here to check your answers.

Use the following table for questions 1 to 5.

Investigating the Product Law of Logarithms
log 12 + log 2 = log 6 + log 4 = log 8 + log 3 =	log 24 =	log m + log n =

1. Use your calculator to evaluate each expression in the first column. Record your results.

2. What do all three expressions in the first column have in common? Describe your observations.

3. Evaluate the expression in the second column. Record your results.

4. How do the values of the expressions in the first column relate to the value of the expression in second column?

5. Based on your observations, make a conjecture for the product law of base 10 logarithms. Record your conjecture in the third column.

Use the following table for questions 6 to 10.

Investigating the Quotient Law of Logarithms
log 12 - log 6 = log 8 - log 4 = log 6 - log 3 =	log 2 =	log m - log n =

6. Use your calculator to evaluate each expression in the first column. Record your results.

7. What do all three expressions in the first column have in common? Describe your observations.

8. Evaluate the expression in the second column. Record your results.

9. How do the values of the expressions in the first column relate to the value of the expression in second column?

10. Based on your observations, make a conjecture for the quotient law of base 10 logarithms. Record your conjecture in the third column.

Use the following table for questions 11 to 15.

Investigating the Power Law of Logarithms
log 2 = log 21 = log 4 = log 22 = log 8 = log 23 = log 16 = log 24 =	1 • log 2 = 2 • log 2 = 3 • log 2 = 4 • log 2 =	log mn =

11. Use your calculator to evaluate each expression in the first column. Record your results.

12. What do all three expressions in the first column have in common? Describe your observations.

13. Evaluate the expression in the second column. Record your results.

14. How do the values of the expressions in the first column relate to the value of the expression in second column?

15. Based on your observations, make a conjecture for the power law of base 10 logarithms. Record your conjecture in the third column.