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Lesson 5-5

Lesson 5-5. Logarithms. Logarithmic functions. Logarithmic functions. The inverse of the exponential function. Logarithmic functions. The inverse of the exponential function. Basic exponential function: f(x) = b x. Logarithmic functions. The inverse of the exponential function.

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Lesson 5-5

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  1. Lesson 5-5 Logarithms

  2. Logarithmic functions

  3. Logarithmic functions The inverse of the exponential function.

  4. Logarithmic functions The inverse of the exponential function. Basic exponential function: f(x) = bx

  5. Logarithmic functions The inverse of the exponential function. Basic exponential function: f(x) = bx

  6. Logarithmic functions The inverse of the exponential function. Basic logarithmic function: f-1(x) = logbx

  7. Logarithmic functions The inverse of the exponential function. Basic logarithmic function: f-1(x) = logbx

  8. Logarithmic functions The inverse of the exponential function. Basic logarithmic function: f-1(x) = logbx Every (x,y)  (y,x)

  9. Logarithmic functions Basic rule for changing exponential equations to logarithmic equations (or vice-versa):

  10. Logarithmic functions Basic rule for changing exponential equations to logarithmic equations (or vice-versa): logbx = a ba = x

  11. Logarithmic functions Basic rule for changing exponential equations to logarithmic equations (or vice-versa): logbx = a ba = x The base of the logarithmic form becomes the base of the exponential form.

  12. Logarithmic functions Basic rule for changing exponential equations to logarithmic equations (or vice-versa): logbx = a ba = x The answer to the log statement becomes the power in the exponential form.

  13. Logarithmic functions Basic rule for changing exponential equations to logarithmic equations (or vice-versa): logbx = a ba = x The number you are to take the log of in the log form, becomes the answer in the exponential form.

  14. Examples:

  15. Examples: log525 = 2 because 52 = 25

  16. Examples: log525 = 2 because 52 = 25 log5125 = 3 because 53 = 125

  17. Examples: log525 = 2 because 52 = 25 log5125 = 3 because 53 = 125 log2(1/8) = - 3 because 2-3 = 1/8

  18. base b exponential function f(x) = bx

  19. base b exponential function f(x) = bx Domain: All reals Range: All positive reals

  20. base b logarithmic function f-1(x) = logb(x)

  21. base b logarithmic function f-1(x) = logb(x) Domain: All positive reals Range: All reals

  22. Types of Logarithms

  23. Types of Logarithms There are two special logarithms that your calculator is programmed for:

  24. Types of Logarithms There are two special logarithms that your calculator is programmed for: log10(x)  called the common logarithm

  25. Types of Logarithms There are two special logarithms that your calculator is programmed for: log10(x)  called the common logarithm For the common logarithm we do not include the subscript 10, so all you will see is: log (x)

  26. Types of Logarithms There are two special logarithms that your calculator is programmed for: So, log10(x)  log (x) = k if 10k = x

  27. Types of Logarithms There are two special logarithms that your calculator is programmed for: loge(x)  called the natural logarithm

  28. Types of Logarithms There are two special logarithms that your calculator is programmed for: loge(x)  called the natural logarithm For the natural logarithm, we do not include the subscript e, so all you will see is: ln (x)

  29. Types of Logarithms There are two special logarithms that your calculator is programmed for: So, loge(x)  ln (x) = k if ek = x

  30. Examples:

  31. Examples: log 6.3 = 0.8 because 100.8 = 6.3

  32. Examples: log 6.3 = 0.8 because 100.8 = 6.3 ln 5 = 1.6 because e1.6 = 5

  33. Example:

  34. Example: Find the value of x to the nearest hundredth.

  35. Example: Find the value of x to the nearest hundredth.

  36. Example: Find the value of x to the nearest hundredth. 10x = 75

  37. Example: Find the value of x to the nearest hundredth. 10x = 75 This transfers to the log statement log 10 75 = x and the calculator will tell you x = 1.88

  38. Example: Find the value of x to the nearest hundredth. ex = 75

  39. Example: Find the value of x to the nearest hundredth. ex = 75 This transfers to the log statement ln 75 = x and the calculator will tell you x = 4.32

  40. Evaluate:

  41. Evaluate:

  42. Evaluate:

  43. Evaluate:

  44. Evaluate:

  45. Solve:

  46. Solve:

  47. Solve:

  48. Solve:

  49. Assignment:Pg. 194C.E. #1 – 9 allW.E. #2 – 14 evens

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