1. Lesson 5-5 Logarithms

2. Logarithmic functions

3. The inverse of the exponential function.

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

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

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

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

8. Logarithmic functions The inverse of the exponential function. Basic logarithmic function: f -1 (x) = log b x 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): log b x = a  b a = x

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

14. Examples:

15. log 5 25 = 2 because 5 2 = 25

16. Examples: log 5 25 = 2 because 5 2 = 25 log 5 125 = 3 because 5 3 = 125

17. Examples: log 5 25 = 2 because 5 2 = 25 log 5 125 = 3 because 5 3 = 125 log 2 (1/8) = - 3 because 2 -3 = 1/8

19. base b exponential function f(x) = b x

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

21. base b logarithmic function f -1 (x) = log b (x)

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

23. Types of Logarithms

24. There are two special logarithms that your calculator is programmed for:

25. Types of Logarithms There are two special logarithms that your calculator is programmed for: log 10 (x)  called the common logarithm

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

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

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

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

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

31. Examples:

32. log 6.3 = 0.8 because 10 0.8 = 6.3

33. Examples: log 6.3 = 0.8 because 10 0.8 = 6.3 ln 5 = 1.6 because e 1.6 = 5

34. Example:

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

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

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

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

39. Example: Find the value of x to the nearest hundredth. e x = 75

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

41. Evaluate:

46. Solve:

50. Assignment: Pg. 194 C.E.  #1 – 9 all W.E.  #2 – 14 evens