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Chapter 4

Chapter 4. 4-4 Properties of logarithms. Objectives. Use properties to simplify logarithmic expressions. Translate between logarithms in any base. Definition of logarithms.

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Chapter 4

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  1. Chapter 4 4-4 Properties of logarithms

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

  3. Definition of logarithms • The logarithmic function is the function , . where b is any number such that is equivalent to    The function is read "log base b of x".     • From the definition above, you can see that every logarithmic equation can be written in an equivalent exponential form and vice versa. • Because logarithms are exponents, you can derive the properties of logarithms from the properties of exponents

  4. Properties of logarithms Remember that to multiply powers with the same base, you add exponents.

  5. Example 1 • Express log64 + log69 as a single logarithm. Simplify.

  6. 1 log 1/3 27 + log 9 Example 2 • Express as a single logarithm. Simplify, if possible. • A) log5625 + log525 • B) 1/3

  7. Student Guided practice • Do problems 1 -3 in your book page 260

  8. Properties of logarithms Remember that to divide powers with the same base, you subtract exponents

  9. Example 3 • Express log5100 – log54 as a single logarithm. Simplify, if possible.

  10. Example 4 • Express log749 – log7 7 as a single logarithm. Simplify, if possible.

  11. Student guided practice • Do problems 4 -6 in your book page 260

  12. Properties of logarithms • Because you can multiply logarithms, you can also take powers of logarithms.

  13. Example 5 • Express as a product. Simplify, if possible. • A. log2326 • B. log8420 • c. log2 ( 1/2 )5

  14. Student guided practice • Do problems 7-10 in your book page 260

  15. Properties of logarithms • Exponential and logarithmic operations undo each other since they are inverse operations.

  16. Example 6 • Simplify each expression. • a. log3311 • b. log381 • c. 5log510

  17. Example 7 • Simplify 2log2(8x)

  18. Student guided practice • Do problems 12-14 in your book page 260

  19. Change of base formula • Most calculators calculate logarithms only in base 10 or basee .You can change a logarithm in one base to a logarithm in another base with the following formula.

  20. Example 8 • Evaluate log328.

  21. Example 9 • Evaluate log927.

  22. Student guided practice • Do problems 15-18 in your book page 260

  23. Properties • 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. • The Richter scale is logarithmic, so an increase of 1 corresponds to a release of 10 times as much energy.

  24. 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? • Solution: • The Richter magnitude of an earthquake, M, is related to the energy released in ergs E given by the formula.

  25. Application#2 • How many times as much energy is released by an earthquake with magnitude of 9.2 by an earthquake with a magnitude of 8?

  26. Homework • Do odd problems from 20-35 in your book page 260

  27. Have a great day!!!

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