1. Evaluating logarithms
A logarithm is an exponent…if there is no base listed,
the common log base is 10.
Evaluate,
Using a
Calculator:

2. Evaluating logarithms
A logarithm is an exponent…if there is no base listed,
the common log base is 10.
Evaluate:

3. Using the calculator- evaluate to the nearest hundredth

4. Using the calculator

5. The natural logarithm: ln
The natural log is base e

6. The natural logarithm: ln
The natural log is base e

7. Logarithmic exponential
Log base value = exponent
Log 2 8 = = 8
Solve for x:
Log 6 x = 3

8. Logarithmic exponential
Log base value = exponent
Log 2 8 = = 8
Solve for x:
Log 6 x = = x
x = 216

9. Change of base law: When not a common logarithm use the change of base

10. Change of base law: When not a common logarithm use the change of base

11. Numerical example-evaluate:

12. Laws of Logarithms

13. Example: from previous slide..
But this is
also true:

14. Example:

15. Examples: expand using log laws:

16. Examples:

17. Single logarithms

18. Single logarithms division, then multiplication:

19. Express as a single logarithm:

20. Express as a single logarithm:

21. Expansion with substitution
Given:
Find:
First expand then substitute

22. Expansion with substitution
First expand then substitute

23. With numbers….
Rewrite the numbers in terms of factors of 4 and 5 only

24. With numbers…. Rewrite the numbers in terms of factors of 4 and 5
And use and find

25. solution
Try page , 50,52

26. Solving log equations: the domain of y = log x is that x>0!
Do Now:1. solve the exponential equation:
2. Solve the logarithmic equation:

27. Solving log equations when the base is a variable:
Recall to raise both change to an
sides to the reciprocal power: exponential equation:

28. Solving log equations…. (combine the last 2 concepts)
Change to an exponential equation
Raise to the reciprocal power.
Check all answers!! (Solutions may be extraneous)

29. Solving log equations….
Change to an exponential equation
Raise to the reciprocal power.

30. More examples: in ex. 1, note that the domain is: x-2>0 so…x>2

31. Solutions: 1. 2.

32. Express as a single log first!

33. Express as a single log first!
-2 is extraneous!

34. When you can’t rewrite using the same base, you can solve by taking a log of both sides
2x = 7
log 2x = log 7
x log 2 = log 7
x = ≈ 2.807

35. Example: 4x = 15

36. Example: 4x = 15 log 4x = log 15 x log 4 = log15 x= log 15/log 4
≈ 1.95

37. Using logarithms to solve exponentials
Here we should “ln” both sides….

38. Using logarithms to solve exponentials

39. Solving with logs-isolate first…

40. Solving with logs-isolate first…

41. Isolate the base term first!
102x +4 = 21

42. Isolate the base term first!
102x +4 = 21
102x = 17
log 102x=log 17

43. Isolate the base term first!
102x +4 = 21
102x = 17
log 102x=log 17
2xlog 10 = log 17
Use ( )!

44. Graphs of exponentials
Growth and decay:
growth decay

45. Compound Formula
Interest rate formula

46. Compound Formula
How long will it take $200 to become $250 at 5% interest rate, compounded quarterly

47. Compound Formula
How long will it take 200 to become 250 at 5% interest rate, compounded quarterly

48. solution
Log both sides and round to the nearest year

49. CONTINUOUS growth:
Ex: population grows continuously at a rate of 2% in Allentown.
If Allentown has 10,000 people today, how many years will it take
To have about 11,000 to the nearest tenth of a year?

50. CONTINUOUS growth:

51. If logbx = logby, then x = y
Solving Log Equations
To solve use the property for logs w/ the same base:
If logbx = logby, then x = y

52. Solve by decompressing
log3(5x-1) = log3(x+7)
Solve by decompressing

53. 5x – 1 = x + 7 5x = x + 8 4x = 8 x = 2 and check
log3(5x-1) = log3(x+7)
5x – 1 = x + 7
5x = x + 8
4x = 8
x = 2 and check
log3(5*2-1) = log3(2+7)
log39 = log39

54. Example:
Solve:

55. Example:
Solve:

56. log5x + log(x+1)=log100
Decompress

57. log5x + log(x+1)=log100 x2 + x - 20 = 0 (subtract 100 and divide by 5)
(5x)(x+1) = (product property)
(5x2 + 5x) = 100
5x2 + 5x-100 = 0
x2 + x - 20 = (subtract 100 and divide by 5)
(x+5)(x-4) = x=-5, x=4
4=x is the only solution

58. another
Solve:

59. another
Solve:

60. One More! log2x + log2(x-7) = 3
Solve and check:

61. One More! log2x + log2(x-7) = 3
log2x(x-7) = 3
log2 (x2- 7x) = 3
2log2(x -7x) = 23
x2 – 7x = 8
x2 – 7x – 8 = 0
(x-8)(x+1)=0
x=8 x= -1 2

62. Graphs of exponentials
Growth and decay:
growth decay

63. Inverse functions Inverse functions are a reflection in y=x
Y=log2x
Y=x
Domain of y=2x is all reals
Domain of y = log2x is