1. Properties of Logarithms Tools for solving logarithmic and exponential equations

2. Let’s review some terms. When we write log 5 125 5 is called the base 125 is called the argument

3. Logarithmic form of 5 2 = 25 is log 5 25 = 2

4. For all the laws a, M and N > 0 a ≠ 1 r is any real

5. Remember ln and log ln is a short cut for log e log means log 10

6. a Easy ones first : log a 1 = 0 since a 0 = 1

7. log 3 1 = ?

8. a log a 1 = 0

9. log 3 1 = 0 a log a 1 = 0

10. ln 1 = ?

11. a log a 1 = 0

12. ln 1 = 0 a log a 1 = 0

13. a Another easy one : log a a = 1 since a 1 = a

14. log 5 5 = ?

15. a log a a = 1

16. log 5 5 = 1 a log a a = 1

17. ln e = ?

18. ln e = log e e = ? ln means log e

19. ln e = log e e = ? a log a a = 1

20. ln e = 1 a log a a = 1

21. a Just a tiny bit harder : log a a r = r since a r = a r

22. ln e 3x = ?

23. ln e 3x = log e e 3x = ? ln means log e

24. ln e 3x = log e e 3x = ?

25. ln e 3x = log e e 3x = 3x

26. log(10 5y ) = ?

27. log means log 10

28. log(10 5y ) = log 10 10 5y = ? log means log 10

29. log(10 5y ) = log 10 10 5y = ?

31. log(10 5y ) = log 10 10 5y = 5y

32. Evidence that it works (not a proof):

34. log(2x) = ?

36. log(2x) = log(2) + log(x)

40. aa Power Rule : log a M r = r log a M Think of it as repeated uses of r times

49. NEVER DO THIS log ( x + y) = log(x) + log(y) (ERROR) WHY is that wrong?  Log laws tell use that log(x) + log(y) = log ( xy) Not log(x + y)

50. Consider 5 = 5 You know that the and the are equal

51. aa So if you knew that : log a M = log a N you would know that M = N

52. aa And vice versa, suppose M = N Then it follows that log a M = log a N

53. ln (x + 7) = ln(10)

54. ln (x + 7) = ln(10) x+7 = 10 ln(M) = ln (N)

55. ln (x + 7) = ln(10) x+7 = 10 x = 3 subtract 7

56. log 3 (x + 5) = log 3 (2x - 4)

57. log 3 (x + 5) = log 3 (2x - 4) log(M) = log(N)

58. log 3 (x + 5) = log 3 (2x - 4) x+5 = 2x - 4 log(M) = log(N)

59. log 3 (x + 5) = log 3 (2x - 4) x+5 = 2x - 4 9 = x oh, this step is easy

60. 3 2x = 5 x

61. If M = N then ln M = ln N 3 2x = 5 x

62. If M = N then ln M = ln N 3 2x = 5 x ln(3 2x ) = ln(5 x )

63. 3 2x = 5 x ln(3 2x ) = ln(5 x )

64. 3 2x = 5 x ln(3 2x ) = ln(5 x ) 2x ln(3 ) = x ln(5)

65. simple algebra 3 2x = 5 x ln(3 2x ) = ln(5 x ) 2x ln(3 ) = x ln(5)

66. simple algebra 3 2x = 5 x ln(3 2x ) = ln(5 x ) 2x ln(3 ) = x ln(5) 2x(ln 3) – x ln(5) = 0

67. factor out x 3 2x = 5 x ln(3 2x ) = ln(5 x ) 2x ln(3 ) = x ln(5) 2x(ln 3) – x ln(5) = 0 x[2ln(3) – ln(5)] = 0

68. Divide out numerical coefficient 3 2x = 5 x ln(3 2x ) = ln(5 x ) 2x ln(3 ) = x ln(5) 2x(ln 3) – x ln(5) = 0 x[2ln(3) – ln(5)] = 0

69. Simplify the fraction 3 2x = 5 x ln(3 2x ) = ln(5 x ) 2x ln(3 ) = x ln(5) 2x(ln 3) – x ln(5) = 0 x[2ln(3) – ln(5)] = 0 =0

70. Change of Base Formula : When you need to approximate log 5 3

72. Here’s one not seen as much as some of the others:

73. Here’s an example