April 12, S5.3q Logarithmic Functions and Graphs

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1 MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 5: Exponential and Logarithmic Functions 5.1 Inverse Functions 5.2 Exponential Functions and Graphs 5.3 Logarithmic Functions and Graphs 5.4 Properties of Logarithmic Functions 5.5 Solving Exponential and Logarithmic Equations 5.6 Applications and Models: Growth and Decay; and Compound Interest Mathematica Interactive Figures are available through Tools for Success, Activities and Projects in CourseCompass. You may access these through CourseCompass or from the Important Links webpage. You must Login to MML to use this link. Graphs of Logarithmic Functions & Inverse Functions: Exponential and Logarithmic Use this program to check logarithmic function graphs. 5.3 Logarithmic Functions and Graphs Find common logarithms and natural logarithms with and without a calculator. Convert between exponential and logarithmic equations. Change logarithmic bases. Graph logarithmic functions. Solve applied problems involving logarithmic functions. Logarithmic Functions These functions are inverses of exponential functions. We can draw the graph of the inverse of an exponential function by interchanging x and y. To Graph: x = 2 y. 1. Choose values for y. 2. Compute values for x. 3. Plot the points and connect them with a smooth curve. * Note that the curve does not touch or cross the y axis. * Note that the curve does NOT touch or cross the y axis. Graph: x = 2 y. The graphs of f(x) = a x and f 1 (x) = loga x for a > 1 and 0 < a < 1 are shown below. If base > 1, Exponential and Logarithmic graphs go UP to the RIGHT. If 0 < base < 1, Exponential and Logarithmic graphs go DOWN to the RIGHT. This curve looks like the graph of y = 2 x reflected across the line y = x, as we would expect for an inverse. The inverse of y = 2 x is x = 2 y. Logarithmic Function, Base a We define y = log a x as that number y such that x = a y, where x > 0 and a is a positive constant other than 1. We read log a x as the logarithm, base a, of x. Nov 15 7:38 AM Finding Certain Logarithms Find each of the following logarithms. a) log 10 10,000 b) log c) log 2 8 d) log 9 3 e) log 6 1 f) log 8 8 a) The exponent to which we raise 10 to obtain 10,000 is 4; thus log10 10,000 = 4. b) The exponent to which we raise 10 to get 0.01 is 2, so log = 2. c) log2 8: 8 = 2 3. The exponent to which we raise 2 to get 8 is 3, so log2 8 = 3. d) log9 3: 3 = 9 1/2. The exponent to which we raise 9 to get 3 is 1/2, so log9 3 = 1/2. e) log6 1: 1 = 6 0. The exponent to which we raise 6 to get 1 is 0, so log6 1 = 0. f) log8 8: 8 = 8 1. The exponent to which we raise 8 to get 8 is 4, so log8 8 = 1. Logarithms loga 1 = 0 and loga a = 1, for any logarithmic base a. A logarithm is an exponent! Convert each of the following to a logarithmic equation. a) 16 = 2 x b) 10 3 = c) e t = 70 Find each of the following common logarithms on a calculator. Round to four decimal places. a) log 645,778 b) log c) log ( 3) Natural Logarithms Logarithms, base e, are called natural logarithms. The abbreviation ln is generally used for natural logarithms. Thus, ln x means loge x. ln 1 = 0 and ln e = 1, for the logarithmic base e. Find each of the following natural logarithms on a calculator. Round to four decimal places. a) ln 645,778b) ln c) log ( 5) d) ln ee) ln 1 1

2 Changing Logarithmic Bases The Change of Base Formula For any logarithmic bases a and b, and any positive number M, Graphs of Logarithmic Functions Method 1 y = log5 x is equivalent to x = 5 y. Select y and compute x. Find log 5 8 using common logarithms. First, we let a = 10, b = 5, and M = 8. Then we substitute into the change of base formula: We can also use base e for a conversion. Find log5 8 using natural logarithms. Method 2 Use a graphing calculator. First change bases. Substituting e for a, 6 for b and 8 for M, we have (continued) Method 3 Calculators which graph inverses automatically. Begin with Y1 = 5 x, the graphs of both Y1 and its inverse Y2 = log 5 x will be drawn. If calculator does not graph inverse function, graph Y2 = (log x) / (log 5) or Y2 = (ln x) / (ln 5). (continued) a) f (x) = ln (x + 3) The graph is a shift 3 units left. The domain is the set of all real numbers greater than 3, ( 3, ). The line x = 3 is the vertical asymptote. Graph each of the following. Describe how each graph can be obtained from the graph of y = ln x. Give the domain and the a) f (x) = ln (x + 3) b) f (x) = 3 ln x c) f (x) = ln (x 1) b) f (x) = 3 ln x The graph is a vertical shrinking of y = ln x, followed by a reflection across the x axis and a translation up 3 units. The domain is the set of all positive real numbers, (0, ). The y axis is the vertical asymptote. (continued) c) f (x) = ln (x 1) The graph is a translation of y = ln x, right 1 unit. The effect of the absolute is to reflect the negative output across the x axis. The domain is the set of all positive real numbers greater than 1, (1, ). The line x =1 is the vertical asymptote. Application In a study by psychologists Bornstein and Bornstein, it was found that the average walking speed w, in feet per second, of a person living in a city of population P, in thousands, is given by the function w(p) = 0.37 ln P a. The population of Hartford, Connecticut, is 124,848. Find the average walking speed of people living in Hartford. b. The population of San Antonio, Texas, is 1,236,249. Find the average walking speed of people living in San Antonio. c. Graph the function. d. A sociologist computes the average walking speed in a city to the approximately 2.0 ft/sec. Use this information to estimate the population of the city. a. Since P is in thousands and 124,848 = thousand, we substitute for P: w( ) = 0.37 ln ft/sec. The average walking speed of people living in Hartford is about 1.8 ft/sec. b. Substitute for P: w( ) = 0.37 ln ft/sec. The average walking speed of people living in San Antonio is about 2.7 ft/sec. 2

3 (continued) c. Graph with a viewing window [0, 600, 0, 4] because inputs are very large and outputs are very small by comparison. 430/2. Make a hand drawn graph of each of the following. Then check your work using a graphing calculator: x = 4 y d. To find the population for which the walking speed is 2.0 ft/sec, we substitute 2.0 for w(p), 2.0 = 0.37 ln P , and solve for P. Use the Intersect method. Graph Y1 = 0.37 ln x and Y2 = /4. Make a hand drawn graph of each of the following. Then check your work using a graphing calculator: x = (4/3) y In a city with an average walking speed of 2.0 ft/sec, the population is about thousand or 194,500. Nov 8 8:27 PM 430/6. Make a hand drawn graph of each of the following. Then check your work using a graphing calculator: y = log4 x Find the following without using a calculator: 430/10. log /15. log 2 (1/4) 430/24. log 10 (8/5) 430/8. Make a hand drawn graph of each of the following. Then check your work using a graphing calculator: f(x) = ln x 430/28. log3 3 2 Nov 8 8:27 PM Nov 8 8:32 PM Convert to a logarithmic equation: 430/ = 1 / 125 Convert to an exponential equation: 430/46. t = log /40. Q t = x 430/48. log 7 = /42. e t = /44. e t = /50. ln 0.38 = /54. ln W 5 = t Nov 8 8:35 PM Nov 8 8:37 PM 3

4 Find each of the following using a calculator. Round to four decimal places: 430/58. log 93, /70. Find the logarithm using common logarithm and change of base formula: log /62. ln /66. ln /74. Find the logarithm using common logarithm and change of base formula: log /68. ln 0 Nov 8 8:39 PM Nov 8 8:40 PM 430/76. Find the logarithm using natural logarithm and change of base formula: log /79. Graph the function and its inverse using the same set of axes. Use any method: f(x) = 3 x, f 1 (x) = log 3 x 430/78. Find the logarithm using natural logarithm and change of base formula: log /80. Graph the function and its inverse using the same set of axes. Use any method: f(x) = log 4 x, f 1 (x) = 4 x Nov 8 8:42 PM Nov 8 8:43 PM 430/81. Graph the function and its inverse using the same set of axes. Use any method: f(x) = log x, f 1 (x) = 10 x 430/84. f(x) = log 3 (x 2) 430/82. Graph the function and its inverse using the same set of axes. Use any method: f(x) = e x, f 1 (x) = ln x 430/88. f(x) = (1 / 2) ln x Nov 8 8:43 PM Nov 8 8:46 PM 4

5 430/90. f(x) = ln (x + 1) 431/95. Walking Speed. Refer to 12. Various cities and their populations are given below. Find the average walking speed in each city. The average walking speed w, in feet per second, of a person living in a city of population P, in thousands, is given by the function w(p) = 0.37 ln P (Source: International Journal of Psychology). a) Seattle, Washington: 608,660 b) Los Angeles, California: 3,792,621 c) Virginia Beach, Virginia: 437, /92. f(x) = (1 / 2)log (x 1) 2 Reference point (2, -2). V.A. x = 1 (x-1) moves basic graph 1 unit to right. -2 moves basic graph down 2 units. (1/2) multiplies y-values by 1/2. Nov 8 8:46 PM 431/96. Forgetting. Students in a computer science class took a final exam and then took equivalent forms of the exam at monthly intervals thereafter. The average score S(t), as a percent, after t months was found to be given by the function S(t) = log (t + 1), t /98. ph of Substances in Chemistry. In chemistry, the ph of a substance is defined as ph = log [H+], where H+ is the hydrogen ion concentration, in moles per liter. Find the ph of each substance. a) Pineapple juice; Hydrogen Ion Concentration 1.6 x 10 4 c) Mouthwash; Hydrogen Ion Concentration 6.3 x 10 7 a) What was the average score when the students initially took the test, t = 0? b) What was the average score after 4 months? after 24 months? c) Graph the function. d) After what time t was the average score 50%? 431/100. Advertising. A model for advertising response is given by the function N(a) = ln a, a 1, where N(a) is the number of units sold when a is the amount spent on advertising, in thousands of dollars. a) How many units were sold after spending $1000 (a = 1) on advertising? b) How many units were sold after spending $5000? c) Graph the function. d) How much would have to be spent in order to sell 2000 units? 5