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Warm Up

Warm Up. Simplify. 1. (2 6 )(2 8 ). 2. (3 –2 )(3 5 ). 3. 4. 5. (7 3 ) 5. Write in exponential form. 6. log x x = 1. 7. 0 = log x 1. Objectives. Use properties to simplify logarithmic expressions. Translate between logarithms in any base.

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Warm Up

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  1. Warm Up Simplify. 1. (26)(28) 2. (3–2)(35) 3. 4. 5. (73)5 Write in exponential form. 6. logxx = 1 7. 0 = logx1

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

  3. 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. Remember that to multiply powers with the same base, you add exponents.

  5. The property in the previous slide can be used in reverse to write a sum of logarithms (exponents) as a single logarithm, which can often be simplified. Helpful Hint Think: log j + log a + log m = log j a m

  6. Check It Out! Example 1a Express as a single logarithm. Simplify, if possible. log5625 + log525

  7. 1 1 3 3 1 log27 + log 9 Check It Out! Example 1b Express as a single logarithm. Simplify, if possible.

  8. Remember that to divide powers with the same base, you subtract exponents Because logarithms are exponents, subtracting logarithms with the same base is the same as finding the logarithms of the quotient with that base.

  9. Caution Just as a5b3 cannot be simplified, logarithms must have the same base to be simplified. The property above can also be used in reverse.

  10. Check It Out! Example 2 Express log749 – log77 as a single logarithm. Simplify, if possible. log749 – log77

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

  12. Check It Out! Example 3 Express as a product. Simplify, if possibly. a. log104 b. log5252

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

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

  15. Check It Out! Example 4 a. Simplify log100.9 b. Simplify 2log2(8x)

  16. 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.

  17. Check It Out! Example 5a Evaluate log927. Method 1 Change to base 10. log927 =

  18. Check It Out! Example 5a Continued Evaluate log927. Method 2 Change to base 3, because both 27 and 9 are powers of 3.

  19. Check It Out! Example 5b Evaluate log816. Method 1 Change to base 10.

  20. Check It Out! Example 5b Continued Evaluate log816. Method 2 Change to base 4, because both 16 and 8 are powers of 2.

  21. 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.

  22. Check It Out! Example 6 How many times as much energy is released by an earthquake with magnitude of 9.2 by an earthquake with a magnitude of 8?

  23. Check It Out! Example 6 Continued

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