1. Logarithms Chapter 5

2. Inverse Functions An American traveling to Europe may find it confusing to find it only being 30 degree weather when they were told to pack shorts and bathing suits. If the function g converts Celsius to Fahrenheit temperatures, then the “inverse of g” would convert Fahrenheit back to Celsius.

3. Inverses The inverse of g: –Switches outputs to inputs instead of inputs to outputs –Inverse of g ( ) undoes g Note: The inverse of a function is not necessarily a function If the inverse of a function is a function it is called invertible.

4. Evaluating Inverse Functions xf(x) 04 116 264 3256 41024 54096 f(4) Warning in, -1 is not an exponent, it is part of function notation to signify an inverse function

5. Using Graphs

6. Function vs Inverse (Tables) xf(x) 45.01 90.1 1351 18010 225100 2701000 xf(x).0145.190 1135 10180 100225 1000270

7. Graphs is the reflection of f across the line x=y for any invertible function f

8. How to plot an inverse function Plot f Chose points on f For each point (a,b), plot points (b,a) Sketch the inverse

9. Example (-7,-1)→(-1,-7) (4,2)→(2,4)

10. Finding an Equation of the Inverse find –substitute s for

11. One-to-one Function - each input has exactly one output One-to-one function - each output originates from exactly one input –All one-to-one functions are invertible –Linear functions with nonzero slopes are one- to-one functions/invertible

12. 5.2 Logarithmic Functions

13. Logarithm Inverse of an exponential function For b > 0, b≠ 1, and a > 0, –the logarithm is the number k such that

14. Logarithm Properties

15. Common Logarithm A logarithm with a base of 10 4 decimal places right 5 decimal places left Only works with 1

16. Logarithmic Function Base, b, is a function that can be put in the form where b > 0 and b ≠ 1

17. Graphing xy 10 21 42 83 xy 01 12 24 38

18. Logarithmic Models In Aug 2011, an earthquake in Virginia had an amplitude of times the reference amplitude. In Jan 2011, California had an earthquake with an amplitude of times the reference amplitude. Find the Richter number of the earthquakes. Find the ratio of the amplitudes.

19. Logarithmic Functions (Cont) Virginia earthquake was 48.5 times greater than the California earthquake

20. 5.3 Solving Equations of Logarithms/Exponents

23. Properties of Logarithms Power Property –For x > 0, b > 0 and b ≠ 1 Equality Property –For positive real numbers a, b, c, b ≠ 1

24. Solving Exponential Equations

25. Solving Using Graphing

26. 5.4 Making Predictions with Exponential Models

27. Change-of-Base Property For a > 0, b > 0, a ≠ 1, b ≠ 1, and x > 0. Since you can change a to any number, using base 10 makes it easy to plug into your calculator for computing logarithms not in base 10

28. Example 1 A person invests $500,000 in an account at 6.5% annual interest, after winning the lottery. Let V=f(t) be the value in dollars of the account after t years. Write an equation

29. What is the V-intercept? –500,000 What does this represent? –This is the initial deposit What is the rate of growth? –1.065 – 1 =.065*100 = 6.5 What does this represent? –The value is growing by 6.5% per year.

30. Find f(5) What does this represent? –The value of the account will be $685,000 after 5 years

31. Find f(t) = 1,000,000 What does this represent? –The value will reach $1,000,000 (double) after 11 years

32. Example 2 World Population in billions is given in the table below. YearPopulation 1930 (0)2.070 1940 (10)2.295 1950 (20)2.500 1960 (30)3.050 1970 (40)3.700 1980 (50)4.454 1990 (60)5.279 2000 (70)6.080

33. Graph a scatterplot of the data Pick two points on the curve –(10, 2.295) (60, 5.279)

34. Find an equation

35. Graph the Line to Check It

36. What is the rate of growth? –1.017 – 1 =.017*100 = 1.7 What does this represent? –The populations is growing by 1.7% per year What is the P-intercept? –1.938 What does this represent? –That the world population (at t = 0) in 1930 was approximately 1.938 billion

37. Predict when the world population will reach 10 billion. 1930 + 97 = 2027 Approximately the year 2027

38. Example 3 Suppose a virus is spreading among a population at an average rate of 2.5% of the population per day. If there are currently 506 people already infected on Oct. 10 th, 2011. On what day will the amount of infected people be doubled t days after Oct. 10 th.

39. Write a formula. Find when the infected population will double. 28 days after Oct. 10 th ~ Nov. 7 th is when the population of infected people will be double

40. When will the number of people infected be tripled. –About 44 days after Oct 10 th ~ Nov 23 rd is when the population of infected people will be tripled

41. What is the I-intercept? –506 What does this number represent? –The number of infected people (at t = 0) on Oct. 10 th, 2011 What is the rate of growth? –1.025 – 1 =.025*100 = 2.5 What does it represent for this situation? –The percent at which the virus is spreading through the population

42. In 2005, a crater was found in a desolate area thought to be formed by a collapsed volcano. If the amount of carbon-14 present in a charcoal sample can be used to determine when the crater formed and the charcoal had 97.32% of the carbon-14 remaining, estimate how long ago it formed. The half-life of carbon-14 is 5730 years. Example 4

43. We know the amount at t = 0 is 100% –S-intercept is (0, 100) We know at 5730 years, the amount will be at 50%. –(5730, 50) Write an equation

44. When was the crater formed? The crater is 226 years old. 2005 – 226 = 1779 The crater was formed in 1779.

45. 5.5 More Logarithm Properties

46. Product Property For x > 0, y > 0, b > 0, and b ≠ 1, –The sum of logarithms is the logarithm of the product of their inputs.

47. Quotient Property For x > 0, y > 0, b > 0, and b ≠ 1, –The difference of two logarithms is the logarithm of the quotient of their inputs

48. Examples

51. Graphing Logarithmic Functions

52. Solving with Graphing (1.76,.27) x ≈ 1.76

53. Warning

54. 5.6 Natural Logarithms

55. Natural Logarithms Logarithm with base e –e ≈ 2.71828182… Note: e is a constant, irrational number, NOT a variable

58. Properties of Natural Logarithms

62. Logarithmic Models A person makes chicken soup. The temperature of the soup decreases by the equation: MinutesTemp (F) 0200 1194 2187 3182 4176 5171

63. Graph a scattergram to check the equation

64. What was the temperature of the soup when it was made (t=0)? –200ºF (y-intercept) If a person waits 6 minutes for the soup to cool before eating, what will the temperature be? –The soup is approximately 166ºF

65. The soup will be “lukewarm” when it reaches a temperature of 98.6ºF, how long will it take to become “lukewarm”? –The soup will be lukewarm after approximately 29 minutes