1. Algebra 2 Logarithmic Functions as Inverses Lesson 7-3

2. Goals Goal To write and evaluate logarithmic expressions. To graph logarithmic functions. Rubric Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems.

3. Vocabulary Logarithm Common Logarithm Logarithmic Scale Logarithmic Function

4. Essential Question Big Idea: Modeling What is a logarithm?

5. Logarithms and Exponentials Inverses How many times would you have to double $1 before you had $8? You could use an exponential equation to model this situation. –1(2 x ) = 8. You may be able to solve this equation by using mental math if you know 2 3 = 8. So you would have to double the dollar 3 times to have $8.

6. Logarithms and Exponentials as Inverses How many times would you have to double $1 before you had $512? You could solve this problem if you could solve 2 x = 8 by using an inverse operation that undoes raising a base to an exponent equation to model this situation. This operation is called finding the logarithm. Logarithm - is the exponent to which a specified base is raised to obtain a given value.

7. For x  0 and 0  a  1, y = log a x if and only if x = a y. The function given by f (x) = log a x is called the logarithmic function with base a. Every logarithmic equation has an equivalent exponential form: y = log a x is equivalent to x = a y A logarithmic function/operation is the inverse function/operation of an exponential function/operation. A logarithm is an exponent! Definition: Logarithmic Function

8. LOGS = EXPONENTS Logarithm - is the exponent to which a specified base is raised to obtain a given value. With this in mind, we can answer questions about the log: This is asking for an exponent. What exponent do you put on the base of 2 to get 16? (2 to the what is 16?) What exponent do you put on the base of 3 to get 1/9? (hint: think negative) What exponent do you put on the base of 4 to get 1? What exponent do you put on the base of 3 to get 3 to the 1/2? (hint: think rational)

9. Your Turn: Find y in each equation. log 2 8 = y  y = 3 log 5 1 = y 25  y = -2 Logarithm - is the exponent to which a specified base is raised to obtain a given value. With this in mind, we can answer questions about the log:

10. Your Turn: Find a in each equation. log a 36 = 2  a = 6 log 8 a = -1/3  a = 1/2 Logarithm - is the exponent to which a specified base is raised to obtain a given value. With this in mind, we can answer questions about the log:

11. You can write an exponential equation as a logarithmic equation and vice versa. Read log b a= x, as “the log base b of a is x.” Notice that the log is the exponent. Reading Math

12. y = log a x if and only if x = a y The logarithmic function to the base a, where a > 0 and a  1 is defined: exponential form logarithmic form Convert to log form: Convert to exponential form: When you convert an exponential to log form, notice that the exponent in the exponential becomes what the log is equal to.

13. Write each exponential equation in logarithmic form. The base of the exponent becomes the base of the logarithm. The exponent is the logarithm. An exponent (or log) can be negative. The log (and the exponent) can be a variable. Exponential Equation Logarithmic Form 3 5 = 243 25 = 5 10 4 = 10,000 6 –1 = a b = c 1 6 1 2 log 3 243 = 5 1 2 log 25 5 = log 10 10,000 = 4 1 6 log 6 = –1 log a c =b Example:

14. Write each exponential equation in logarithmic form. The base of the exponent becomes the base of the logarithm. The exponent of the logarithm. The log (and the exponent) can be a variable. Exponential Equation Logarithmic Form 9 2 = 81 3 3 = 27 x 0 = 1(x ≠ 0) a. b. c. log 9 81 = 2 log 3 27 = 3 log x 1 = 0 Your Turn:

15. Write each logarithmic form in exponential equation. The base of the logarithm becomes the base of the power. The logarithm is the exponent. A logarithm can be a negative number. Any nonzero base to the zero power is 1. Logarithmic Form Exponential Equation log 9 9 = 1 log 2 512 = 9 log 8 2 = log 4 = –2 log b 1 = 0 1 16 1 3 9 1 = 9 2 9 = 512 1 3 8 = 2 1 16 4 –2 = b 0 = 1 Example:

16. Write each logarithmic form in exponential equation. The base of the logarithm becomes the base of the power. The logarithm is the exponent. An logarithm can be negative. Logarithmic Form Exponential Equation log 10 10 = 1 log 12 144 = 2 log 8 = –3 1 2 10 1 = 10 12 2 = 144 1 2 –3 = 8 Your Turn:

17. A logarithm is an exponent, so the rules for exponents also apply to logarithms. You may have noticed the following properties in the last example. Special Properties of Logarithms

30. Definition: Common Logarithm - a logarithm with base 10. If no base is written for a logarithm, the base is assumed to be 10. For example, log 5 = log 10 5.

31. Evaluate by using mental math. The log is the exponent. Think: What power of 10 is 0.01? log 0.01 10 ? = 0.01 10 –2 = 0.01 log 0.01 = –2 Example:

32. Evaluate by using mental math. The log is the exponent. Think: What power of 5 is 125? log 5 125 5 ? = 125 5 3 = 125 log 5 125 = 3 Example:

33. Evaluate by using mental math. The log is the exponent. Think: What power of 5 is ? log 5 1 5 5? =5? = 1 5 5 –1 = 1 5 log 5 = –1 1 5 1 5 Example:

34. Evaluate by using mental math. The log is the exponent. Think: What power of 10 is 0.01? log 0.00001 10 ? = 0.00001 10 –5 = 0.01 log 0.00001 = –5 Your Turn:

35. Evaluate by using mental math. The log is the exponent. Think: What power of 25 is 0.04? log 25 0.04 25 ? = 0.04 25 –1 = 0.04 log 25 0.04 = –1 Your Turn:

36. Logarithmic Scale Many measurements of physical phenomena have such a wide range of values that the reported measurements are logarithms (exponents) of the values, not the values themselves. Logarithmic Scale – when you use the logarithm of a quantity instead of the quantity. The Richter scale is a logarithmic scale. It gives logarithmic measurements of earthquake magnitude.

37. Earthquakes

38. The largest earthquake ever recorded was in Chile. It measured 8.9 on the Richter Scale. This measures the magnitude of a tremor (how powerful it is) using an instrument called a seismograph. On the Richter Scale, magnitude is expressed in whole numbers and decimal fractions. Although the Richter Scale has no upper limit, the largest known shocks have had magnitudes in the 8.8 to 8.9 range. It is a logarithmic scale which means that a size ‘6’ on the Richter Scale is 10 times larger than a size ’5’ and 100 times larger than a size ‘4’. 1 2 3 4 6 5 9 8 7 10 Richter Scale The Richter Scale How can we measure earthquakes?

39. The Japanese earthquake in Kobe (September 1995) measured 7.2 on the Richter Scale. The Greek earthquake (June 1995) measured 6.2 on the Richter Scale. How many times greater was the Japanese earthquake? 1 2 3 4 6 5 9 8 7 10 Richter Scale Example: This can be done by using the formula, which compares the intensity levels of earthquakes where I is the intensity level determined by a seismograph, and M is the magnitude on the Richter scale.

40. Example: Continued The Japanese earthquake:7.2 The Greek earthquake: 6.2 The Japanese earthquake was 10 times as strong as the Greek earthquake.

41. Earthquake struck on holiday honouring Dr Martin Luther King….57 dead Fires burned out of control last night after a devastating earthquake measuring 6.6 on the Richter Scale hit LA. Over fifty people have been killed including fourteen people trapped in a collapsed bock of flats, near the epicentre, in the district of Northridge. Reports suggest that over one thousand people are injured and the city is at a standstill. Freeways have buckled, trains have been derailed and the airport is closed. The earthquake struck before dawn and was felt over a wide area. The quake was felt as far away as Las Vegas, 125 miles away to the east! Los Angeles earthquake (1/94) Turkey earthquake (8/99) Izmit buildings substandard The earthquake that hit Turkey last month has resulted in an estimated death toll of between 30,000 and 40,000. The earthquake that measured 7.4 on the Richter Scale, struck at 3am. It had an epicentre approximately 11 km to the south east of Izmit and it was felt as far as 320km away. Turkey received international help to rescue the thousands trapped in collapsed buildings. At least 20,000 buildings collapsed or suffered heavy damage. The buildings which collapsed were mainly between 6 and 8 stories high and had been built in the last few years. Although new buildings in earthquake areas are supposed to follow the ‘Uniform Buildings Code’ (California), many of these buildings were poorly constructed in concrete and had unreinforced masonry walls. Your Turn: How many times more intense was the Turkey earthquake than the LA earthquake?

42. Continued The Turkey earthquake was 6.31 times as strong as the LA earthquake.

43. LOGARITHMIC FUNCTIONS

44. Essential Question Big Idea: Modeling What is the relationship between exponential functions and logarithmic functions?

45. Logs and exponentials are inverse functions of each other so let’s see what we can tell about the graphs of logs based on what we learned about the graphs of exponentials. For functions and their inverses, x’s and y’s trade places. So anything that was true about x’s or the domain of a function, will be true about y’s or the range of the inverse function and vice versa. Let’s look at the characteristics of the graphs of exponentials then and see what this tells us about the graphs of their inverse functions which are logarithms. Logarithms and Exponentials as Inverse Functions

46. Characteristics about the Graph of an Exponential Function b > 1 1. Domain is all real numbers 2. Range is positive real numbers 3. There are no x intercepts because there is no x value that you can put in the function to make it = 0 4. The y intercept is always (0,1) because b 0 = 1 5. The graph is always increasing 6. The x-axis (where y = 0) is a horizontal asymptote for x  -  Characteristics about the Graph of a Log Function b > 1 1. Range is all real numbers 2. Domain is positive real numbers 3. There are no y intercepts 4. The x intercept is always (1,0) (x’s and y’s trade places) 5. The graph is always increasing 6. The y-axis (where x = 0) is a vertical asymptote

47. Because logarithms are the inverses of exponents, the inverse of an exponential function, such as y = 2 x, is a logarithmic function, such as y = log 2 x. You may notice that the domain and range of each function are switched. The domain of y = 2 x is all real numbers (R), and the range is {y|y > 0}. The domain of y = log 2 x is {x|x > 0}, and the range is all real numbers (R). Logarithmic Function

48. Use the x-values {–2, –1, 0, 1, 2}. Graph the function and its inverse. Describe the domain and range of the inverse function. f(x) = 1.25 x Graph f(x) = 1.25 x by using a table of values. 1 f(x) = 1.25 x 210–1–2x 0.640.81.251.5625 Example: Graphing

49. To graph the inverse, the logarithmic function g(x) = log 1.25 x, by using a table of values (switch the x and y values). 210–1–2 g(x) = log 1.25 x 1.5625 1.25 1 0.80.64 x The domain of g(x) is {x|x > 0}, and the range is R. Example: Continued

50. x–2–1012 f(x) =( ) x 421 Graph f(x) = x by using a table of values. 1 2 1 2 1 2 1 4 f(x) = x 1 2 Use the x-values {–2, –1, 0, 1, 2}. Graph the function and its inverse. Describe the domain and range of the inverse function. Example:

51. The domain of g(x) is {x|x > 0}, and the range is R. To graph the inverse, the logarithmic function g(x) = log x, by using a table of values (switch the x and y values). 1 2 1 2 1 4 1 2 x421 g(x) =log x –2–1012 Example: Continued 1 2

52. x–2–1123 f(x) = x 16 9 4 3 3 4 9 27 64 3 4 Use x = –2, –1, 1, 2, and 3 to graph. Then graph its inverse. Describe the domain and range of the inverse function. Graph by using a table of values. Your Turn:

53. The domain of f –1 (x) is {x|x > 0}, and the range is R. To graph the inverse, g(x) = log x, by using a table of values. 3 4 x g(x) = log x –2–1123 16 9 4 3 3 4 9 27 64 3 4 Your Turn:

54. Graphs of Logarithmic Functions The graphs of logarithmic functions are similar for different values of b. f(x) = log b x (b  1) 3. x-intercept (1, 0) 5. increasing 6. continuous 7. one-to-one 8. reflection of y = b x in y = x 1. domain 2. range 4. vertical asymptote Graph of f (x) = log b x (b  1) x y y = x y = log 2 x y = b x domain range y-axis vertical asymptote x-intercept (1, 0)

55. Graphs of Logarithmic Functions Typical shape for graphs where b > 1 (includes base e and base 10 graphs). Typical shape for graphs where 0 < b < 1.

56. The key is used to evaluate logarithms in base 10. is used to find 10 x, the inverse of log. Helpful Hint

58. Transformations Logarithmic Functions

62. Transformation of functions apply to log functions just like they apply to all other functions so let’s try a couple. up 2 left 1 Reflect about x axis

63. Essential Question Big Idea: Modeling What is a logarithm? A logarithm y is the exponent to which base b must be raised to get a number x. So, log b x = y if and only if b y = x.

64. Essential Question Big Idea: Modeling What is the relationship between exponential functions and logarithmic functions? Because the exponential function y = b x is one-to-one, its inverse is a function. The logarithmic function y = log b x is the inverse of the exponential function y = b x. The graph of these functions are reflections of each other over the line y = x.

65. Assignment Section 7-3, Pg 485 – 487; #1 – 11 all, 12 – 38 even, 42 – 46 even, 54 – 64 even.