LOGARITHMS AND THEIR PROPERTIES
Definition of a logarithm: If and is a constant , then if and only if . In the equation is referred to as the logarithm, is the base , and is the argument.
The notation is read “the logarithm (or log) base of .” The definition of a logarithm indicates that a logarithm is an exponent. is the logarithmic form of is the exponential form of
Examples of changes between logarithmic and exponential forms:
Write each equation in its exponential form.
a. b. c.
Solution:
Use the definition if and only if .
Logarithms are exponents . a. b. a
. a c. a
Base
Write the following in its logarithmic form:
Solution:
Use . a
Exponent
Base
Equality of Exponents Theorem: If is positive real number such that , then .
Example of Evaluating a Logarithmic Equation:
Evaluate:
Solution :
a c a Thus, by Equality of Exponents,
PROPERTIES OF LOGARITHMS:
If b, a, and c are positive real numbers, , and n is a real number, then:
1. Product : 5. 2. Quotient: 6. Inverse 1: 3. Power: 7. Inverse 2: 8. One-to-One : 4. a 9. Change of Base:
Examples – Rewriting Logarithmic Expressions Using Logarithmic Properties: Use the properties of logarithms to rewrite each expression as a single logarithm:
b. a. Solution:
b. a. Power Property Power Property Product Property Quotient Property
Use the properties of logarithms to express the following logarithms in terms of logarithms of . a
a. b.
Solution:
a. Product Property b. Power Property Quotient Property Quotient Property Product Property Power Property Other Logarithmic Definitions: • Definition of Common Logarithm : Logarithms with a base of 10 are called common logarithms. It is customary to write . a • Definition of Natural Logarithm: Logarithms with the base of are called natural logarithms. It is customary to write . a
PRACTICE PROBLEMS
Evaluate:
1. 2. . 3. 4. 5. 6. . . Rewrite into logarithms:
7. 8. 9. . Evaluate without a calculator:
10. 12. 11. Use the change of base formula to evaluate the logarithms: (Round to 3 decimal places.)
13. 15. 14. 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. 22. 17. 20. 18. 21. Write each expression as a single logarithmic quantity:
23. 26. 29. 24. 27. 28. 25. Using properties of logarithms find the following values if:
. . . 30. 31. 32. 33. 34. Write the exponential equation in logarithmic form:
35. 36. Write the logarithmic equation in exponential form:
37. 38. Evaluate the following logarithms without a calculator:
39. 45. 42. 40. 46. 43. 41. 44.
Evaluate the following logarithms for the given values of : 47. a. b. c. . 48. a. b. c. . . 49. a. c. b. 50. a. c. b. 51. a. b. c. . 52. a. b. c. . Use the change of base formula to evaluate the following logarithms: (Round to 3 decimal places.) 53. 54. 55. 56. . Approximate the following logarithms given that and : . . 57. 60. 58. 61. 59. 62. Use the properties of logarithms to expand the expression:
63. 69. 66. 64. 70. 67. 65. 68. Use the properties of logarithms to condense the expression:
75. 79. 71. 76. 80. 72. 77. 73. 74. 78. 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. 85. 84. 82. 86.
Practice Problems Answers Note: Remember that all variables that represent an argument of a logarithm must be greater than 0.
1. 33. 562 . . 66. 2. 34. . 67. 3. 35. . 68. 4. 36. 69. 5. 37. 6. 70. . 38. 7. 39. 8. 71. 40. 9. . 72. 41. 10. 73. 42. 2 11. 74. 43. 12. 2 44. 75. 13. 565 . 45. 76. 14. 1 46. 15. 380 . 77. 47. a. b. c. 16. . 48. a. b. c. 78. 17. .
49. a. b. c. 79. 18. . . 50. a. b. c. 80. 19. . . 51. a. b. c. 81. False. 20. . 52. a. b. c. . 82. False. 21. 53. . 83. True 22. 54. . 84. True 55. 132 23. . 85. True 56. 1.159 24. 86. True 57. 7959 . 25. 58. 556645 . 26. 59. 0.43068 27. 60. 0.25193 28. 61. 02931 . 62. 11329 29. . 63. 30. . 64. 31. . 65. 32. .