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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

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