Use properties to simplify logarithmic expressions. Translate between logarithms in any base .

1. Objectives Use properties to simplify logarithmic expressions. Translate between logarithms in any base.

2. The logarithmic function for pH that you saw in the previous lessons, pH =–log[H+], can also be expressed in exponential form, as 10–pH= [H+]. Because logarithms are exponents, you can derive the properties of logarithms from the properties of exponents

4. Example 1: Adding Logarithms Express log64 + log69 as a single logarithm. Simplify. log64 + log69 To add the logarithms, multiply the numbers. log6 (4  9) log6 36 Simplify. Think: 6? = 36. 2

5. Check It Out! Example 1a Express as a single logarithm. Simplify, if possible. log5625 + log525 To add the logarithms, multiply the numbers. log5 (625 • 25) log5 15,625 Simplify. 6 Think: 5? = 15625

6. 1 1 1 1 3 3 3 1 9 log27 + log 9 log(27 • ) log 3 1 1 3 3 Think: ? = 3 Check It Out! Example 1b Express as a single logarithm. Simplify, if possible. To add the logarithms, multiply the numbers. Simplify. –1

7. The property above can also be used in reverse.

8. Example 2: Subtracting Logarithms Express log5100 – log54 as a single logarithm. Simplify, if possible. log5100 – log54 To subtract the logarithms, divide the numbers. log5(100 ÷4) Simplify. log525 2 Think: 5? = 25.

9. Check It Out! Example 2 Express log749 – log77 as a single logarithm. Simplify, if possible. log749 – log77 To subtract the logarithms, divide the numbers log7(49 ÷ 7) log77 Simplify. 1 Think: 7? = 7.

10. Because you can multiply logarithms, you can also take powers of logarithms.

11. Because 8 = 4, log84 = . 20() = 40 2 2 2 3 3 3 3 Example 3: Simplifying Logarithms with Exponents Express as a product. Simplify, if possible. A. log2326 B. log8420 6log232 20log84 Because 25 = 32, log232 = 5. 6(5) = 30

12. Check It Out! Example 3 Express as a product. Simplify, if possibly. a. log104 b. log5252 4log10 2log525 Because 101 = 10, log10 = 1. Because 52 = 25, log525 = 2. 4(1) = 4 2(2) = 4

13. 5log2 ( ) Because 2–1 = , log2 = –1. 1 1 2 2 1 1 2 2 Check It Out! Example 3 Express as a product. Simplify, if possibly. c. log2 ( )5 5(–1) = –5

14. Exponential and logarithmic operations undo each other since they are inverse operations.

15. Example 4: Recognizing Inverses Simplify each expression. c. 5log510 a. log3311 b. log381 log3311 log33  3  3 3 5log510 log334 10 11 4

16. Check It Out! Example 4 a. Simplify log100.9 b. Simplify 2log2(8x) log 100.9 2log2(8x) 8x 0.9

17. Most calculators calculate logarithms only in base 10 or base e (see Lesson 7-6). You can change a logarithm in one base to a logarithm in another base with the following formula.

18. log8 log328 = log32 ≈ 0.903 1.51 Example 5: Changing the Base of a Logarithm Evaluate log328. Method 1 Change to base 10 Use a calculator. Divide. ≈ 0.6

19. log28 log328 = 3 log232 = 5 Example 5 Continued Evaluate log328. Method 2 Change to base 2, because both 32 and 8 are powers of 2. Use a calculator. = 0.6

20. log27 log927 = log9 ≈ 1.431 0.954 Check It Out! Example 5a Evaluate log927. Method 1 Change to base 10. Use a calculator. ≈ 1.5 Divide.

21. log327 log927 = 3 log39 = 2 Check It Out! Example 5a Continued Evaluate log927. Method 2 Change to base 3, because both 27 and 9 are powers of 3. Use a calculator. = 1.5

22. log16 Log816 = log8 ≈ 1.204 0.903 Check It Out! Example 5b Evaluate log816. Method 1 Change to base 10. Use a calculator. Divide. ≈ 1.3

23. log416 log816 = 2 log48 = 1.5 Check It Out! Example 5b Continued Evaluate log816. Method 2 Change to base 4, because both 16 and 8 are powers of 2. Use a calculator. = 1.3

24. Helpful Hint The Richter scale is logarithmic, so an increase of 1 corresponds to a release of 10 times as much energy. 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.

25. Example 6: Geology Application The tsunami that devastated parts of Asia in December 2004 was spawned by an earthquake with magnitude 9.3 How many times as much energy did this earthquake release compared to the 6.9-magnitude earthquake that struck San Francisco in1989? The Richter magnitude of an earthquake, M, is related to the energy released in ergs E given by the formula. Substitute 9.3 for M.

26. Multiply both sides by . æ ö E 13.95 = log ç ÷ è 11.8 10 ø 3 2 Example 6 Continued Simplify. Apply the Quotient Property of Logarithms. Apply the Inverse Properties of Logarithms and Exponents.

27. Example 6 Continued Given the definition of a logarithm, the logarithm is the exponent. Use a calculator to evaluate. The magnitude of the tsunami was 5.6 1025 ergs.

28. Multiply both sides by . 3 2 Example 6 Continued Substitute 6.9 for M. Simplify. Apply the Quotient Property of Logarithms.

29. 5.6  1025 The tsunami released = 4000 times as much energy as the earthquake in San Francisco. 1.4  1022 Example 6 Continued Apply the Inverse Properties of Logarithms and Exponents. Given the definition of a logarithm, the logarithm is the exponent. Use a calculator to evaluate. The magnitude of the San Francisco earthquake was 1.4  1022 ergs.