1. Tuesday Bellwork Pair-Share your homework from last night We will review this briefly Remember, benchmark tomorrow

2. This week: Monday: Logarithmic Functions and Their Graphs Tuesday: “ “ cont. Wednesday: Benchmark Thursday: Properties of Logarithms Friday: Review/Quiz OR Solving Exponential with Logarithms

3. The standard F.BF.5. Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

4. LOGARITHMIC FUNCTIONS AND THEIR GRAPHS

5. PROPERTIES OF LOGARITHMS Section 3.3

6. Consider This is a one-to-one function, therefore it has an inverse. The inverse is called a logarithm function. Example: Two raised to what power is 16? The most commonly used bases for logs are 10: and e : is called the natural log function. is called the common log function.

7. Definition of Logarithmic Function b > 0; b  1 Logarithmic Form Exponential Form y = log b xx = b y

8. The log to the base “ b ” of “ x ” is the exponent to which “ b ” must be raised to obtain “ x ” y = log 10 x y = log e x  x = 10 y  x = e y Section I on HW

9. Change from Logarithmic To Exponential Form Log 2 8 = 3  8 = 2 3  5 = 25 ½ Log 25 5 = ½

10. Change from Exponential To Form Logarithmic 49 = 7 2  log 7 49 = 2 1/5 = 5 –1  log 5 (1/5)= -1 Section II on HW

11. Using the Definition of log! 1. log 3 x = 4 2. log 10 0.01 = x 3. log x 49 = 2 x = 3 4 = 81 0.01 = 10 x 10 -2 = 10 x x = -2 49 = x 2 x=7 Section III on HW

12. YOU use the definition of log: Write each equation in its equivalent exponential form. a. 2 = log 5 xb. 3 = log b 64c. log 3 7 = y SolutionWith the fact that y = log b x means b y = x, c. log 3 7 = y or y = log 3 7 means 3 y = 7. a. 2 = log 5 x means 5 2 = x. Logarithms are exponents. b. 3 = log b 64 means b 3 = 64. Logarithms are exponents. Section III on HW

13. How to evaluate expressions: Pre-Calc Cookbook: 1. Set expression equal to y. 2. Identify the ‘b’ & ‘x’ 3. Use the formula to convert to exponential form. 4. Make common bases (if not already) 5. Solve for y. Similar to HW section IV

14. Example: Pre-Calc Cookbook: 1.Set expression equal to y. 2.Identify the ‘b’ & ‘x’ 3.Use the formula to convert to exponential form. 4.Make common bases (if not already) 5.Solve for y.

15. a. log 2 16b. log 3 9 c. log 25 5 Solution log 25 5 = 1/2 because 25 1/2 = 525 to what power is 5?c. log 25 5 log 3 9 = 2 because 3 2 = 93 to what power is 9?b. log 3 9 log 2 16 = 4 because 2 4 = 162 to what power is 16?a. log 2 16 Logarithmic Expression Evaluated Question Needed for Evaluation Logarithmic Expression YOU Evaluate the expressions:

16. Properties of Logarithmic Functions If b, M, and N are positive real numbers, b  1, and p and x are real numbers, then: Log 15 1 = 0 Log 10 10 = 1 Log 5 5 x = x 3 log x = x 3 15 0 = 1 10 1 = 10 5 x = 5 x

17. Properties of Common Logarithms General PropertiesCommon Logarithms 1. log b 1 = 01. log 1 = 0 2. log b b = 12. log 10 = 1 3. log b b x = 03. log 10 x = x 4. b log b x = x 4. 10 log x = x

18. Examples of Logarithmic Properties log 4 4 = 1 log 8 1 = 0 3 log 3 6 = 6 log 5 5 3 = 3 2 log 2 7 = 7

19. Properties of Natural Logarithms General PropertiesNatural Logarithms 1. log b 1 = 01. ln 1 = 0 2. log b b = 12. ln e = 1 3. log b b x = 03. ln e x = x 4. b log b x = x 4. e ln x = x

20. Examples of Natural Logarithmic Properties e log e 6 = e ln 6 = 6 log e e 3 = 3

21. Standard Based Questions: Use the inverse properties to simplify: Section V on HW

22. Tuesday Independent Practice Logarithm Functions HW3: Complete Sections I, II, III, IV, & V.

23. WEDNESDAY BENCHMARK!!!

24. Thursday

25. Thursday Bellwork  Answer the following questions on a separate piece of paper that you can turn in?  What did you think of the benchmark?  What do we need more practice on? What do we have mastery of?

26. Characteristics of the Graphs of Logarithmic Functions of the Form f (x) = log b x  The x-intercept is 1. There is no y-intercept.  The y-axis is a vertical asymptote. (x = 0)  If 0 1, the function is increasing.  The graph is smooth and continuous. It has no sharp corners or edges. -2 6 2345 5 4 3 2 -2 6 f (x) = log b x b>1 -2 6 2345 5 4 3 2 -2 6 f (x) = log b x 0<b<1

27. Since logs and exponentials are inverses the domain and range switch!…the x values and y values are exchanged…

29. Graph and find the domain of the following functions. y = ln x x y -2 0 1 2 3 4.5 cannot take the ln of a (-) number or 0 0 ln 2 =.693 ln 3 = 1.098 ln 4 = 1.386 ln.5 = -.693 D: x > 0

30. f xy = 2 x –3 1 8 –2 1 4 –1 1 2 01 12 24 38 f x = 2y 1 8 –3 1 4 –2 1 2 –1 10 21 42 83 Orderedpairs reversed y x y 510–5 5 10 –5 f -1 x = 2 y or y = log 2 x f y = 2 x y =x DOMAIN of = (– ,  ) = RANGE of RANGE of f = (0,  ) = DOMAIN of Logarithmic Function with Base 2 f f -1

31. Using Calculator to Evaluate: ln(10)  > Calculate  > ‘ctrl’ then ‘e x ’ => ln( )  > ’10’ => ln(10)  > ‘ enter’ => ln(10)  > ‘menu’  > ’2: Number’  > ‘1: Convert to Decimal’ => Ans>Decimal  > ‘enter’ => 2.30259

32. We Evaluate: ln(12.4)  > Calculate  > ‘ctrl’ then ‘e x ’ => ln( )  > ’10’ => ln(12.4)  > ‘ enter’ => ln(12.4)  > ‘menu’  > ’2: Number’  > ‘1: Convert to Decimal’ => Ans>Decimal  > ‘enter’ => 2.5177

33. YOU Calculator to Evaluate: 1. ln(45) = 2. ln(0.234) = 3. ln(-3.45) = 1.= 3.80666 1.= -1.45243 2.= non-real number Similar to section VI in HW

34. Homework:  Complete ALL Logarithm Functions HW 3

35. FRIDAY BELLWORK

36. Copyright © Cengage Learning. All rights reserved. 3.3 Properties of Logarithms

37. What You Should Learn Rewrite logarithms with different bases. Use properties of logarithms to evaluate or rewrite logarithmic expressions. Use properties of logarithms to expand or condense logarithmic expressions. Use logarithmic functions to model and solve real-life problems.

38. Properties of Logarithms

40. Example 1 – Using Properties of Logarithms Write each logarithm in terms of ln 2 and ln 3. a.ln 6 b. ln Solution: a. ln 6 = ln(2  3) = ln 2 + ln 3 b. ln = ln 2 – ln 27 = ln 2 – ln 3 3 = ln 2 – 3 ln 3 Rewrite 6 as 2  3. Product Property Quotient Property Rewrite 27 as 3 3 Power Property

41. Rewriting Logarithmic Expressions

42. The properties of logarithms are useful for rewriting logarithmic expressions in forms that simplify the operations of algebra. This is true because they convert complicated products, quotients, and exponential forms into simpler sums, differences, and products, respectively.

43. Example 2 – Expanding Logarithmic Expressions Use the properties of logarithms to expand each expression. a. log 4 5x 3 y b. ln Solution: a. log 4 5x 3 y = log 4 5 + log 4 x 3 + log 4 y = log 4 5 + 3 log 4 x + log 4 y Product Property Power Property

44. Example 2 – Solution Rewrite radical using rational exponent. Power Property Quotient Property cont’d

45. Rewriting Logarithmic Expressions In Example 5, the properties of logarithms were used to expand logarithmic expressions. In Example 6, this procedure is reversed and the properties of logarithms are used to condense logarithmic expressions.

46. Example 3 – Condensing Logarithmic Expressions Use the properties of logarithms to condense each expression. a. log 10 x + 3 log 10 (x + 1) b. 2ln(x + 2) – lnx c. [log 2 x + log 2 (x – 4)]

47. Example 3 – Solution a. log 10 x + 3 log 10 (x + 1) = log 10 x 1/2 + log 10 (x + 1) 3 b. 2 ln(x + 2) – ln x = ln(x + 2) 2 – ln x Power Property Product Property Quotient Property Power Property

48. Example 3 – Solution c. [log 2 x + log 2 (x – 4)] = {log 2 [x(x – 4)]} = log 2 [x(x – 4)] 1/3 cont’d Power Property Product Property Rewrite with a radical.

49. Homework: Properties of Logs HW4

50. Monday, March 23, 2015 F.BF.5. Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

51. 3.4 Exponential and Logarithmic Equations One-to-One Properties Inverse Property

52. One-to-One Properties If x 6 = x y, then 6 = y If ln a = ln b, then a = b

53. Inverse Property Given e x = 8; solve for x Take the natural log of each side. ln e x = ln 8 Pull the exponent in front x ( ln e) = ln 8 (since ln e = 1) x = ln 8

54. Solve for x 3 x = 64 take the natural log of both sides ln 3 x = ln 64 x( ln 3) = ln 64 x = ln 64=3.7855.. ln 3

55. Solve for x e x – 8 = 70

56. Solve for x e x – 8 = 70 e x = 78 ln e x = ln 78 x = ln 78 x = 4.3567..

57. Solve for a ( ¼ ) a = 64

58. Solve for K Log 5 K = - 3

59. Solve for x 2 x – 3 = 32

60. Solve for x

61. e 2.724x = 29

62. Solve for a ln a + ln ( a + 3) = 1 Will need the quadratic formula

63. Solve for x one more time e 2x – e x – 12 = 0 factor

64. Solve for x one more time e 2x – e x – 12 = 0 factor (e x – 4)(e x + 3 ) = 0 So e x – 4 = 0or e x + 3 = 0 e x = 4e x = - 3 x = ln 4x = ln -3

65. Homework: Solving Exponential Equations with Logarithms HW5