LOGARITHMS AND THEIR PROPERTIES Definition of a logarithm: If and is a constant , then if and only if 4 . ͬ Ƙ 0 ͖ ʚ͖ ƕ 1ʛ ͭ Ɣ log ͬ ͖ Ɣ ͬ In the equation is referred to as the logarithm, is the base , and is the argument. ͭ Ɣ log ͬ , ͭ ͖ ͬ The notation is read “the logarithm (or log) base of .” The definition of a logarithm indicates that a logarithm log ͬ ͖ ͬ is an exponent. is the logarithmic form of 4 ͭ Ɣ log ͬ ͖ Ɣ ͬ 4 is the exponential form of ͖ Ɣ ͬ ͭ Ɣ log ͬ Examples of changes between logarithmic and exponential forms: Write each equation in its exponential form. a. b. c. 2 Ɣ log ͫ ͬ 3 Ɣ log ͥͤ ʚͬ ƍ 8ʛ log ͩ 125 Ɣ ͬ Solution: Use the definition if and only if 4 . ͭ Ɣ log ͬ ͖ Ɣ ͬ Logarithms are exponents ͧ . a. b. 3 Ɣ log ͥͤ ʚͬ ƍ 8ʛ if and only if 10 Ɣ ʚͬ ƍ 8ʛ ͦ 3 . 2 Ɣ log ͫ ͬ if and only if 7 Ɣ ͬ c. log ͩ 125 Ɣ ͬ if and only if 5 Ɣ 125 Base Write the following in its logarithmic form: ͥ⁄ ͦ ͬ Ɣ 25 Solution: Use 4 . ͬ Ɣ ͖ if and only if ͭ Ɣ log ͬ Exponent ͥ⁄ͦ 1 ͬ Ɣ 25 ͚͝ ͕͘͢ ͣͭ͢͠ ͚͝ Ɣ log ͦͩ ͬ Base 2 Equality of Exponents Theorem: If is positive real number such that 3 4, then . ͖ ʚ͖ ƕ 1ʛ ͖ Ɣ ͖ ͬ Ɣ ͭ Example of Evaluating a Logarithmic Equation: Evaluate: log ͦ 32 Ɣ ͬ Solution : 3 log ͦ 32 Ɣ ͬ if and only if 2 Ɣ 32 ͩ 3 ͩ Since 32 Ɣ 2 , we have 2 Ɣ 2 Thus, by Equality of Exponents, ͬ Ɣ 5 PROPERTIES OF LOGARITHMS: If b, a, and c are positive real numbers, , and n is a real number, then: ¸ ƕ ̊ 1. Product : 5. log ʚ͕ · ͗ʛ Ɣ log ͕ ƍ log ͗ log ͖ Ɣ 1 2. Quotient: 6. Inverse 1: ) log ͖ Ɣ ͢ log Ɣ log ͕ Ǝ log ͗ ΚΝΕ ) 3. Power: ) 7. Inverse 2: ĕ log ͕ Ɣ ͢ · log ͕ ͖ Ɣ ͢, ͢ Ƙ 0 8. One-to-One : 4. log 1 Ɣ 0 log ͕ Ɣ log ͗ if and only if ͕ Ɣ ͗ 9. Change of Base: ΚΝΕ Ė ΚΝΕ ΚΜ log ͕ Ɣ Ɣ Ɣ ΚΝΕ Ė ΚΝΕ ΚΜ Examples – Rewriting Logarithmic Expressions Using Logarithmic Properties: Use the properties of logarithms to rewrite each expression as a single logarithm: ͥ b. a. 4 log ʚͬ ƍ 2ʛ Ǝ 3 log ʚͬ Ǝ 5ʛ 2 log ͬ ƍ ͦ log ʚͬ ƍ 4ʛ Solution: ͥ b. a. 4 log ʚͬ ƍ 2ʛ Ǝ 3 log ʚͬ Ǝ 5ʛ 2 log ͬ ƍ ͦ log ʚͬ ƍ 4ʛ ͨ ͧ Power Property ͦ ͥ⁄ ͦ Power Property Ɣ log ʚͬ ƍ 2ʛ Ǝ log ʚͬ Ǝ 5ʛ Ɣ log ͬ ƍ log ʚͬ ƍ 4ʛ ʚ3ͮͦʛx ͦ ͥ⁄ ͦ Product Property Quotient Property Ɣ log ƫͬ ʚͬ ƍ 4ʛ Ư Ɣ log ʚ3ͯͩʛw Use the properties of logarithms to express the following logarithms in terms of logarithms of . ͬ, ͭ, and ͮ a. ͦ 3v 4 log ʚͬͭ ʛ b. √ log 5y Solution: a. ͦ ͦ Product Property ͦ log ʚͬͭ ʛ Ɣ log ͬ ƍ log ͭ b. ͬ ǭͭ log ͩ Power Property ͮ Ɣ log ͬ ƍ 2 log ͭ ͦ ͩ Quotient Property Ɣ log Ƴͬ ǭͭƷ Ǝ log ͮ ͦ ͩ Quotient Property Ɣ log Ƴͬ ǭͭƷ Ǝ log ͮ 2 5 Product Property Ɣ log ͬ ƍ log ǭͭ Ǝ log ͮ u Power Property Ɣ 2 log ͬ ƍ v log ͭ Ǝ 5 log ͮ Other Logarithmic Definitions: • Definition of Common Logarithm : Logarithms with a base of 10 are called common logarithms. It is customary to write . log ͥͤ ͬ as logͬ • Definition of Natural Logarithm: Logarithms with the base of are called natural logarithms. It is customary to write . ͙ log ͬ as ln ͬ PRACTICE PROBLEMS Evaluate: 1. ͧ 2. ͧ.ͦ 3. ͨͤͤ 4. 5. 6. 0.6√ ͙ ʚ1.005 ʛ log ͨ 64 ln 1 ln √7 Rewrite into logarithms: 7. ͨ 8. 9. ͨ 2 Ɣ 16 √64 Ɣ 8 ͙ Ɣ 54.60 Evaluate without a calculator: 10. ͥ 12. ͯͦ log ͩ 25 11. ln ͙ log ͧ ͬͥ Use the change of base formula to evaluate the logarithms: (Round to 3 decimal places.) 13. ͥ 15. log ͫ 3 14. log ͥͩ 42 log ͦ ͦ Use the properties of logarithms to rewrite each expression into lowest terms (i.e. expand the logarithms to a sum or a difference): 16. 19. ͦ ͧ3 log 10ͬ log ͨ 4ͬ 22. √ ln ͫ 17. 34 20. ln 5 log ͧ √ͬ Ǝ 2 y v 18. 3x 21. 3 5 ln w log 5v 4 Write each expression as a single logarithmic quantity: 23. 26. ͥ log 7 Ǝ log ͬ log ͦ 5 ƍ log ͦ ͬ Ǝ log ͦ 3 29. ͦ log ͩ 7 Ǝ 2 log ͩ ͬ 24. 27. 3lnͬƍ2lnͭƎ4lnͮ 1 ƍ 3 log ͨ ͬ ͧ ͧ 28. 25. ͪ ͬ 2 ln 8 ƍ 5 ln ͬ ͦ ln ͬ Ǝ ͨ ln ͬ Using properties of logarithms find the following values if: log 3 Ɣ 0.562 log 2 Ɣ 0.356 log 7 Ɣ 0.872 30. 31. 32. ͥ 33. ͦ 34. log 18 log 28 log 3͖ log 1 √ log ͦͥ Write the exponential equation in logarithmic form: 35. ͧ 36. ͧ⁄ ͦ 4 Ɣ 64 25 Ɣ 125 Write the logarithmic equation in exponential form: 37. ͥ ln ͙ Ɣ 1 38. log ͧ ͭ Ɣ Ǝ2 Evaluate the following logarithms without a calculator: 39. ͥ 45. log 1000 42. ln 1 log ͨ ͥͪ 40. 46. ͯͧ log ͭ 3 43. ͫ ln ͙ ln ͙ ͥ 41. ͥ log ͧ ͭ 44. log Evaluate the following logarithms for the given values of : ͬ 47. ͚ʚͬʛ Ɣ log ͧ ͬ a. b. c. ͬ Ɣ 1 ͬ Ɣ 27 ͬ Ɣ 0.5 48. ͛ʚͬʛ Ɣ log ͬ a. b. c. ͬ Ɣ 0.01 ͬ Ɣ 0.1 ͬ Ɣ 30 49. ͚ʚͬʛ Ɣ ln ͬ a. ͥ c. ͬ Ɣ ͙ b. ͬ Ɣ 10 ͬ Ɣ ͧ 50. ͜ʚͬʛ Ɣ ln ͬ a. ͦ ͩ c. ͬ Ɣ ͙ b. ͬ Ɣ 1200 ͬ Ɣ ͨ 51. ͧ3 ͛ʚͬʛ Ɣ ln ͙ a. b. c. ͬ Ɣ Ǝ2 ͬ Ɣ 0 ͬ Ɣ 7.5 52. ͚ʚͬʛ Ɣ log ͦ √ͬ a. b. c. ͬ Ɣ 4 ͬ Ɣ 64 ͬ Ɣ 5.2 Use the change of base formula to evaluate the following logarithms: (Round to 3 decimal places.) 53. 54. 55. 56. log ͨ 9 log ͥ⁄ ͦ 5 log ͥͦ 200 log ͧ 0.28 Approximate the following logarithms given that and : log ͩ 2 Ɩ 0.43068 log ͩ 3 Ɩ 0.68261 57. ͦ log ͩ 18 60. log ͩ ͧ 58. log ͩ √6 61. ͦ⁄ ͧ log ͩʚ12 ʛ ͥ 59. 62. ͦ log ͩ ͦ log ͩʚ5 · 6ʛ Use the properties of logarithms to expand the expression: 63. ͨ 69. ͩ log ͨ 6ͬ 66. w 3 ln ƫ√2ͬʚͬ ƍ 3ʛ Ư ln ǯ 64. ͯͧ ͩ log 2ͬ 70. v√ 67. 3ͮͦ log ͧ y 65. ln log ͩ √ͬ ƍ 2 3ͯͦ 68. ͦ ln ͬʚͬ Ǝ 3ʛ Use the properties of logarithms to condense the expression: ͦ 75. 79. 71. Ǝ2ʚln 2ͬ Ǝ ln 3ʛ 3lnͬƍ4lnͭƍlnͮ Ǝ ͧ ln 3ͭ 76. 80. 72. 4ʚ1 ƍ ln ͬ ƍ ln ͬʛ ln ʚͬ ƍ 4ʛ Ǝ 3 ln ͬ Ǝ ln ͭ 5 log ͦ ͭ 77. 73. ͦ 4ʞlog ͦ ͟ Ǝ log ͦʚ͟ Ǝ ͨʛʟ log ͬ 16ͬ ƍ log ͬ 2ͬ ͥ 74. 78. log ͨ 6ͬ Ǝ log ͨ 10 ͧ ʚlog ͬ ͕ ƍ 2 log ͬ ͖ʛ True or False? Use the properties of logarithms to determine whether the equation is true or false. If false, state why or give an example to show that it is false. 81. 83. ͦ3 ͥͪ log ͦ 4ͬ Ɣ 2log ͦ ͬ log 10 Ɣ 2ͬ 85. log ͨ 3 Ɣ 2 Ǝ log ͨ ͬ ΚΜ ͩ3 ͥ 84. ΚΜ / 82. ͙ Ɣ ͨ 86. ͪ ΚΜ ͥͤ3 Ɣ ln ͦ 6lnͬ ƍ6lnͭ Ɣ ln ʚͬͭ ʛ Practice Problems Answers Note: Remember that all variables that represent an argument of a logarithm must be greater than 0. 1. 33. 562 ͥ 0.413 2. 66. ͧ ʚln ͬ Ǝ ln 5ʛ 2. 34. 24.533 0 67. ln ʚͬ ƍ 2ʛ Ǝ ln ʚͬ Ǝ 2ʛ 3. 35. 7.352 log ͨ 64 Ɣ 3 68. lnͬ ƍ 2ln ʚͬ Ǝ 3ʛ 4. 36. ͧ 3 log ͦͩ 125 Ɣ 69. ͥ ͦ ʚ ʛ ʚ ʛ 5. ͦ ln 2 ƍ ln ͬ ƍ 5 ln ͬ ƍ 3 0 37. ͥ ͙ Ɣ ͙ ͥ 6. 70. 2 log ͧ ͕ ƍ ͦ log ͧ ͖ Ǝ log ͧ ͗ Ǝ 0.973 38. ͯͦ ͥ 7. 3 Ɣ ͭ 5 log ͧ ͘ log ͦ 16 Ɣ 4 39. 3 ⁄ 8. ͥ 71. ͥ ͦ ͧ log ͪͨ 8 Ɣ ͦ 40.