Notes 6-3

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Logarithm Functions

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Notes 6-3

  1. 1. Section 6-3 L o g a r i t h m s
  2. 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. 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. 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. 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. 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. 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. 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. 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. 10. Definition of Logarithm
  11. 11. Definition of Logarithm Let b > 0 and b ≠ 1. Then y is the logarithm of x to the base b, written:
  12. 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. 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. 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. 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. 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. 17. Example 1 Evaluate. 1 a. log 6 6 b. log 6 36 5 c. log 6 36
  18. 18. Example 1 Evaluate. 1 a. log 6 6 b. log 6 36 5 c. log 6 36 −1
  19. 19. Example 1 Evaluate. 1 a. log 6 6 b. log 6 36 5 c. log 6 36 −1 Why?
  20. 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. 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. 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. 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. 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. 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. 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. 27. Example 2 Evaluate. log 9 243
  28. 28. Example 2 Evaluate. log 9 243 9 = 81 2
  29. 29. Example 2 Evaluate. log 9 243 9 = 81 2 9 = 729 3
  30. 30. Example 2 Evaluate. log 9 243 9 = 81 2 x is somewhere in between 9 = 729 3
  31. 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. 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. 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. 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. 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. 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. 37. Common Logarithms
  38. 38. Common Logarithms Logarithms with a base of 10
  39. 39. Common Logarithms Logarithms with a base of 10 You will see this one on your calculator
  40. 40. Example 3 Solve to the nearest hundredth. 10 = 73 y
  41. 41. Example 3 Solve to the nearest hundredth. 10 = 73 y Ok, let’s rewrite this as a logarithm.
  42. 42. Example 3 Solve to the nearest hundredth. 10 = 73 y Ok, let’s rewrite this as a logarithm. log 73 = y
  43. 43. Example 3 Solve to the nearest hundredth. 10 = 73 y Ok, let’s rewrite this as a logarithm. log 73 = y
  44. 44. Example 3 Solve to the nearest hundredth. 10 = 73 y Ok, let’s rewrite this as a logarithm. log 73 = y
  45. 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. 46. Example 4 Solve log t = 2.9 to the nearest tenth.
  47. 47. Example 4 Solve log t = 2.9 to the nearest tenth. Rewrite as a power.
  48. 48. Example 4 Solve log t = 2.9 to the nearest tenth. Rewrite as a power. 10 2.9 =t
  49. 49. Example 4 Solve log t = 2.9 to the nearest tenth. Rewrite as a power. 10 2.9 =t t ≈ 794.3
  50. 50. Properties of Logarithms
  51. 51. Properties of Logarithms Domain is the set of positive real numbers.
  52. 52. Properties of Logarithms Domain is the set of positive real numbers. Range is the set of all real numbers.
  53. 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. 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. 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. 56. Properties of Logarithms
  57. 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. 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. 59. Homework
  60. 60. Homework p. 387 #1 - 26

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