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

Objectives. Use properties to simplify logarithmic expressions. Translate between logarithms in any base. Because logarithms are exponents, you can derive the properties of logarithms from the properties of exponents.

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

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

  2. Because logarithms are exponents, you can derive the properties of logarithms from the properties of exponents Remember that to multiply powers with the same base, you add exponents.

  3. 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+ logm = log jam

  4. 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. 1 1 3 3 1 log27 + log 9 Express as a single logarithm. Simplify, if possible. log5625 + log525

  6. 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.

  7. 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.

  8. 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. 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() = 2 40 2 2 3 3 3 3 Express as a product. Simplify, if possible. A. log2326 B. log8420 6log232 20log84 6(5) = 30 Because 25 = 32, log232 = 5.

  12. 5log2 ( ) Because 2–1 = , log2 = –1. 1 1 2 2 1 1 2 2 Express as a product. Simplify, if possibly. a. log104 b. log5252 c. log2 ( )5 4log10 2log525 4(1) = 4 2(2) = 4 5(–1) = –5 Because 52 = 25, log525 = 2. Because 101 = 10, log10 = 1.

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

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

  15. Simplify each expression. a. Simplify log100.9 b. Simplify 2log2(8x) log 100.9 2log2(8x) 8x 0.9

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

  18. log27 log16 log927 = Log816 = log9 log8 ≈ ≈ 1.204 1.431 0.954 0.903 Changing the Base of a Logarithm Evaluate log927. Evaluate log816. Change to base 10. Change to base 10. ≈ 1.5 ≈ 1.3

  19. 7 6 Lesson Quiz: Part I Express each as a single logarithm. 1. log69 + log624 log6216 = 3 log327 = 3 2. log3108 – log34 Simplify. 3. log2810,000 30,000 4. log44x –1 x – 1 5. 10log125 125 6. log64128

  20. 8. log 10 1 2 Lesson Quiz: Part II Use a calculator to find each logarithm to the nearest thousandth. 2.727 7. log320 –3.322

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