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Notes 6-3

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Logarithm Functions

Logarithm Functions

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  • 1. Section 6-3 L o g a r i t h m s
  • 2. Warm-up Solve without a calculator. a. 10 = .0001 a b. 10 = .01 a c. 10 = 1 a d. 10 = 10 a e. 10 = 100 a f. 10 = 100,000,000,000 a g. 10 = 0 a
  • 3. Warm-up Solve without a calculator. a. 10 = .0001 a b. 10 = .01 a c. 10 = 1 a a = −4 d. 10 = 10 a e. 10 = 100 a f. 10 = 100,000,000,000 a g. 10 = 0 a
  • 4. Warm-up Solve without a calculator. a. 10 = .0001 a b. 10 = .01 a c. 10 = 1 a a = −4 a = −2 d. 10 = 10 a e. 10 = 100 a f. 10 = 100,000,000,000 a g. 10 = 0 a
  • 5. Warm-up Solve without a calculator. a. 10 = .0001 a b. 10 = .01 a c. 10 = 1 a a = −4 a = −2 a=0 d. 10 = 10 a e. 10 = 100 a f. 10 = 100,000,000,000 a g. 10 = 0 a
  • 6. Warm-up Solve without a calculator. a. 10 = .0001 a b. 10 = .01 a c. 10 = 1 a a = −4 a = −2 a=0 d. 10 = 10 a e. 10 = 100 a a= 1 2 f. 10 = 100,000,000,000 a g. 10 = 0 a
  • 7. Warm-up Solve without a calculator. a. 10 = .0001 a b. 10 = .01 a c. 10 = 1 a a = −4 a = −2 a=0 d. 10 = 10 a e. 10 = 100 a a= 1 2 a=2 f. 10 = 100,000,000,000 a g. 10 = 0 a
  • 8. Warm-up Solve without a calculator. a. 10 = .0001 a b. 10 = .01 a c. 10 = 1 a a = −4 a = −2 a=0 d. 10 = 10 a e. 10 = 100 a a= 1 2 a=2 f. 10 = 100,000,000,000 a g. 10 = 0 a a = 11
  • 9. Warm-up Solve without a calculator. a. 10 = .0001 a b. 10 = .01 a c. 10 = 1 a a = −4 a = −2 a=0 d. 10 = 10 a e. 10 = 100 a a= 1 2 a=2 f. 10 = 100,000,000,000 a g. 10 = 0 a a = 11 No solution
  • 10. Definition of Logarithm
  • 11. Definition of Logarithm Let b > 0 and b ≠ 1. Then y is the logarithm of x to the base b, written:
  • 12. Definition of Logarithm Let b > 0 and b ≠ 1. Then y is the logarithm of x to the base b, written: y = log b x IFF b = x y
  • 13. Definition of Logarithm Let b > 0 and b ≠ 1. Then y is the logarithm of x to the base b, written: y = log b x IFF b = x y What does this mean?
  • 14. Definition of Logarithm Let b > 0 and b ≠ 1. Then y is the logarithm of x to the base b, written: y = log b x IFF b = x y What does this mean? y = log b x IFF b = x y
  • 15. Definition of Logarithm Let b > 0 and b ≠ 1. Then y is the logarithm of x to the base b, written: y = log b x IFF b = x y What does this mean? y = log b x IFF b = x y Base
  • 16. Definition of Logarithm Let b > 0 and b ≠ 1. Then y is the logarithm of x to the base b, written: y = log b x IFF b = x y What does this mean? y = log b x IFF b = x y Base Exponent
  • 17. Example 1 Evaluate. 1 a. log 6 6 b. log 6 36 5 c. log 6 36
  • 18. Example 1 Evaluate. 1 a. log 6 6 b. log 6 36 5 c. log 6 36 −1
  • 19. Example 1 Evaluate. 1 a. log 6 6 b. log 6 36 5 c. log 6 36 −1 Why?
  • 20. Example 1 Evaluate. 1 a. log 6 6 b. log 6 36 5 c. log 6 36 −1 Why? −1 6 = 1 6
  • 21. Example 1 Evaluate. 1 a. log 6 6 b. log 6 36 5 c. log 6 36 −1 2 Why? −1 6 = 1 6
  • 22. Example 1 Evaluate. 1 a. log 6 6 b. log 6 36 5 c. log 6 36 −1 2 Why? Why? −1 6 = 1 6
  • 23. Example 1 Evaluate. 1 a. log 6 6 b. log 6 36 5 c. log 6 36 −1 2 Why? Why? −1 6 = 1 6 6 = 36 2
  • 24. Example 1 Evaluate. 1 a. log 6 6 b. log 6 36 5 c. log 6 36 2 −1 2 5 Why? Why? −1 6 = 1 6 6 = 36 2
  • 25. Example 1 Evaluate. 1 a. log 6 6 b. log 6 36 5 c. log 6 36 2 −1 2 5 Why? Why? Why? −1 6 = 1 6 6 = 36 2
  • 26. Example 1 Evaluate. 1 a. log 6 6 b. log 6 36 5 c. log 6 36 2 −1 2 5 Why? Why? Why? −1 6 = 1 2 6 6 = 36 2 6 = 36 = 6 5 5 5 2
  • 27. Example 2 Evaluate. log 9 243
  • 28. Example 2 Evaluate. log 9 243 9 = 81 2
  • 29. Example 2 Evaluate. log 9 243 9 = 81 2 9 = 729 3
  • 30. Example 2 Evaluate. log 9 243 9 = 81 2 x is somewhere in between 9 = 729 3
  • 31. Example 2 Evaluate. log 9 243 9 = 81 2 x is somewhere in between 9 = 729 3 What do we know about 243?
  • 32. Example 2 Evaluate. log 9 243 9 = 81 2 x is somewhere in between 9 = 729 3 What do we know about 243? 5 243 = 3
  • 33. Example 2 Evaluate. log 9 243 9 = 81 2 x is somewhere in between 9 = 729 3 What do we know about 243? 1 5 243 = 3 = 9 2
  • 34. Example 2 Evaluate. log 9 243 9 = 81 2 x is somewhere in between 9 = 729 3 What do we know about 243? 1 5 243 = 3 = 9 2 Ok, what does that mean?
  • 35. Example 2 Evaluate. log 9 243 9 = 81 2 x is somewhere in between 9 = 729 3 What do we know about 243? 1 5 243 = 3 = 9 2 Ok, what does that mean? (9 ) = 243 1 5 2
  • 36. Example 2 Evaluate. log 9 243 9 = 81 2 x is somewhere in between 9 = 729 3 What do we know about 243? 1 5 243 = 3 = 9 2 Ok, what does that mean? (9 ) = 243 1 5 2 log 9 243 = 5 2
  • 37. Common Logarithms
  • 38. Common Logarithms Logarithms with a base of 10
  • 39. Common Logarithms Logarithms with a base of 10 You will see this one on your calculator
  • 40. Example 3 Solve to the nearest hundredth. 10 = 73 y
  • 41. Example 3 Solve to the nearest hundredth. 10 = 73 y Ok, let’s rewrite this as a logarithm.
  • 42. Example 3 Solve to the nearest hundredth. 10 = 73 y Ok, let’s rewrite this as a logarithm. log 73 = y
  • 43. Example 3 Solve to the nearest hundredth. 10 = 73 y Ok, let’s rewrite this as a logarithm. log 73 = y
  • 44. Example 3 Solve to the nearest hundredth. 10 = 73 y Ok, let’s rewrite this as a logarithm. log 73 = y
  • 45. Example 3 Solve to the nearest hundredth. 10 = 73 y Ok, let’s rewrite this as a logarithm. log 73 = y y ≈ 1.86
  • 46. Example 4 Solve log t = 2.9 to the nearest tenth.
  • 47. Example 4 Solve log t = 2.9 to the nearest tenth. Rewrite as a power.
  • 48. Example 4 Solve log t = 2.9 to the nearest tenth. Rewrite as a power. 10 2.9 =t
  • 49. Example 4 Solve log t = 2.9 to the nearest tenth. Rewrite as a power. 10 2.9 =t t ≈ 794.3
  • 50. Properties of Logarithms
  • 51. Properties of Logarithms Domain is the set of positive real numbers.
  • 52. Properties of Logarithms Domain is the set of positive real numbers. Range is the set of all real numbers.
  • 53. Properties of Logarithms Domain is the set of positive real numbers. Range is the set of all real numbers. (1, 0) will be on the graph; logb1 = 0.
  • 54. Properties of Logarithms Domain is the set of positive real numbers. Range is the set of all real numbers. (1, 0) will be on the graph; logb1 = 0. The function is strictly increasing.
  • 55. Properties of Logarithms Domain is the set of positive real numbers. Range is the set of all real numbers. (1, 0) will be on the graph; logb1 = 0. The function is strictly increasing. As x increases, y has no bound.
  • 56. Properties of Logarithms
  • 57. Properties of Logarithms As x gets smaller and approaches 0, the values of the function are negative with larger absolute values. That means when x is between 0 and 1, the exponent will be negative.
  • 58. Properties of Logarithms As x gets smaller and approaches 0, the values of the function are negative with larger absolute values. That means when x is between 0 and 1, the exponent will be negative. The y-axis is an asymptote.
  • 59. Homework
  • 60. Homework p. 387 #1 - 26