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Logarithmic Functions

PreCalculus NYOS Charter School Quarter 4 “If we did all the things we were capable of doing, we would literally astound ourselves .” ~ Thomas Edison. Logarithmic Functions. Logarithmic Functions. The logarithmic function y = log a x, where a > 0 and a ≠ 1,

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Logarithmic Functions

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  1. PreCalculusNYOS Charter SchoolQuarter 4“If we did all the things we were capable of doing, we would literally astound ourselves.” ~ Thomas Edison Logarithmic Functions

  2. Logarithmic Functions • The logarithmic function y = loga x, where a > 0 and a ≠ 1, is the inverseof the exponential function y = ax. y = loga x iff x = ay

  3. Logarithmic Functions Example: Write in exponential form. log3 9 = 2

  4. Logarithmic Functions Example: Write in exponential form. log3 9 = 2

  5. Logarithmic Functions Example: Write in exponential form. log8 2 =

  6. Logarithmic Functions Example: Write in exponential form. log8 2 =

  7. Logarithmic Functions Example: Write in exponential form. log125 25 =

  8. Logarithmic Functions Example: Write in exponential form. log125 25 =

  9. Logarithmic Functions Example: Write in logarithmic form.

  10. Logarithmic Functions Example: Write in logarithmic form. log4 64 =

  11. Logarithmic Functions Example: Write in logarithmic form.

  12. Logarithmic Functions Example: Write in logarithmic form. log3=

  13. Logarithmic Functions Example: Evaluate log7. y = log7 y = -2

  14. Logarithmic Functions Example: Evaluate log5. y = log5

  15. Logarithmic Functions Example: Evaluate log5. y = log5 y = -3

  16. Logarithmic Functions Properties of Logarithms

  17. Logarithmic Functions Example: Expand log5 9x = log5 9 + log5 x

  18. Logarithmic Functions Example: Expand logx12y

  19. Logarithmic Functions Example: Expand logx12y = logx12 + logxy

  20. Logarithmic Functions Properties of Logarithms

  21. Logarithmic Functions Example: Expand log5 9/x = log5 9 - log5 x

  22. Logarithmic Functions Example: Expand logx12/y

  23. Logarithmic Functions Example: Expand logx12/y = logx12 - logxy

  24. Logarithmic Functions Properties of Logarithms

  25. Logarithmic Functions Example: Simplify log5 9x = x log5 9

  26. Logarithmic Functions Properties of Logarithms

  27. Logarithmic Functions Example: Simplify. log5 9 = log5 x 9 = x

  28. Logarithmic Functions Example: Solve for x. log5 16 = log5 2x 16 = 2x 8 = x

  29. Logarithmic Functions Properties of Logarithms

  30. Logarithmic Functions Example: Simplify. log5 5 1

  31. Logarithmic Functions Example: Simplify. log87 87 1

  32. Logarithmic Functions Example: Simplify. log87 1 0

  33. Logarithmic Functions Example: Simplify. log48 1 0

  34. Logarithmic Functions Example: Solve. log8 48 – log8 w = log8 6

  35. Logarithmic Functions Example: Solve. log8 48 – log8 w = log8 6 log8(48/w) = log86 48/w = 6 w = 8

  36. Logarithmic Functions Example: Solve. log10= x

  37. Logarithmic Functions Example: Solve. log10= x log10= x x =

  38. Logarithmic Functions • If a, b, and n are positive numbers and neither a nor b is 1, then the following is called the change of base formula:

  39. Logarithmic Functions Example: Rewrite with a base of 2. log6 5 =

  40. Logarithmic Functions Example: Combine. = log11 15

  41. Logarithmic Functions • Natural logarithms have base e. ln 5

  42. Logarithmic Functions Example: Convert log6 254 to a natural logarithm and evaluate. log6 254 = ≈ 3.09

  43. Logarithmic Functions Example: Convert log5 43 to a natural logarithm and evaluate. log5 43

  44. Logarithmic Functions Example: Convert log5 43 to a natural logarithm and evaluate. log5 43 = ≈ 2.34

  45. Logarithmic Functions Example: Solve using natural logs. 2x = 27 log2 27 = x = x x ≈ 4.75

  46. Logarithmic Functions Example: Solve. 9x-4 = 7.13

  47. Logarithmic Functions Example: Solve. 9x-4 = 7.13 log9 7.13 = x - 4 + 4 = x ≈ 4.89

  48. Logarithmic Functions Example: Solve. 6x+2 = 14 The variable is in the exponent. Take the log of both sides. ln6x+2 = ln 14

  49. Logarithmic Functions Example: Solve. 6x+2 = 14 ln6x+2 = ln 14 (x + 2) ln 6 = ln 14 x + 2 = x ≈ -.53

  50. Logarithmic Functions Example: Solve. 2x-5 = 11

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