1. 7.4 P ROPERTIES OF L OGARITHMS Use properties to simplify logarithmic expressions. Translate between logarithms in any base. Objectives Why are we learning this? Logarithmic scales are useful for measuring quantities that have a very wide range of values, such as the intensity (loudness) of a sound or the energy released by an earthquake. Warm Up 2. (3 –2 )(3 5 ) 1. (2 6 )(2 8 ) 3. 4. 5. (7 3 ) 5 Simplify.

2. Remember that to multiply powers with the same base, you add exponents. Check that is true using the given!: Think: log j + log a + log m = log jam Helpful Hint

3. Express log 6 4 + log 6 9 as a single logarithm. Simplify. Example 1: Adding Logarithms log 6 4 + log 6 9 log 27 + log 1 3 1 3 1 9

4. Remember that to divide powers with the same base, you subtract exponents Check that is true using the given!:

5. Express log 5 100 – log54 as a single logarithm. Simplify, if possible. Example 2: Subtracting Logarithms log 5 100 – log54 log 7 49 – log77

6. Illustrate this law with:

7. Express as a product. Simplify, if possible. Example 3: Simplifying Logarithms with Exponents A. log 2 32 6 b. log 5 25 2

8. Exponential and log operations undo each other since they are inverse operations. Prove that these are true:

9. Example 4: Recognizing Inverses Simplify each expression. b. log 3 81 c. 5 log 5 10 a. log 3 3 11 log 3 3 11 11 5 log 5 10 10

10. Example 5: Changing the Base of a Logarithm Evaluate log 32 8. Method 1 Change to base 10 log 32 8 = log8 log32 0.903 1.51 ≈ ≈ 0.6 Use a calculator. Divide.