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 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

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. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

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 = 2y. 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. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example Graph: x = 2y. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example (continued) This curve looks like the graph of y = 2x reflected across the line y = x, as we would expect for an inverse. The inverse of y = 2x is x = 2y. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Logarithmic Function, Base a We define y = loga x as that number y such that x = ay, where x > 0 and a is a positive constant other than 1. We read loga x as “the logarithm, base a, of x.” Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Finding Certain Logarithms - Example Find each of the following logarithms. a) log10 10,000 b) log10 0.01 c) log2 8 d) log9 3 e) log6 1 f) log8 8 Solution: a) The exponent to which we raise 10 to obtain 10,000 is 4; thus log10 10,000 = 4. b) We have The exponent to which we raise 10 to get 0.01 is –2, so log10 0.01 = –2. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example (continued) c) log2 8: The exponent to which we raise 2 to get 8 is 3, so log2 8 = 3. d) log9 3: The exponent to which we raise 9 to get 3 is 1/2; thus log9 3 = 1/2. e) log6 1: 1 = 60. The exponent to which we raise 6 to get 1 is 0, so log6 1 = 0. f) log8 8: 8 = 81. The exponent to which we raise 8 to get 8 is 4, so log8 8 = 1. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Logarithms loga 1 = 0 and loga a = 1, for any logarithmic base a. A logarithm is an exponent! Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example Convert each of the following to a logarithmic equation. a) 16 = 2x b) 10–3 = 0.001 c) et = 70 The exponent is the logarithm. a) 16 = 2x log216 = x The base remains the same. Solution: b) 10–3 = 0.001 g log10 0.001 = –3 c) et = 70 g log e 70 = t Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example Convert each of the following to an exponential equation. a) log 2 32= 5 b) log a Q= 8 c) x = log t M The logarithm is the exponent. a) log 2 32 = 5 25 = 32 The base remains the same. Solution: b) log a Q = 8 g a8 = Q c) x = log t M g tx = M Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example Find each of the following common logarithms on a calculator. Round to four decimal places. a) log 645,778 b) log 0.0000239 c) log (3) Solution: Function Value Readout Rounded a) log 645,778 5.8101 b) log 0.0000239 –4.6216 c) log (–3) Does not exist. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 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. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example Find each of the following natural logarithms on a calculator. Round to four decimal places. a) ln 645,778 b) ln 0.0000239 c) log (5) d) ln e e) ln 1 Solution: Function Value Readout Rounded a) ln 645,778 13.3782 b) ln 0.0000239 –10.6416 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example (continued) Solution: Function Value Readout Rounded c) ln (–5) Does not exist. b) ln e 1 c) ln 1 0 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Changing Logarithmic Bases The Change-of-Base Formula For any logarithmic bases a and b, and any positive number M, Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example Find log5 8 using common logarithms. Solution: First, we let a = 10, b = 5, and M = 8. Then we substitute into the change-of-base formula: Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example We can also use base e for a conversion. Find log5 8 using natural logarithms. Solution: Substituting e for a, 6 for b and 8 for M, we have Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Graphs of Logarithmic Functions - Example Graph: y = f (x) = log5 x. Solution: y = log5 x is equivalent to x = 5y. Select y and compute x. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example Graph each of the following. Describe how each graph can be obtained from the graph of y = ln x. Give the domain and the vertical asymptote of each function. a) f (x) = ln (x + 3) b) f (x) = c) f (x) = |ln (x – 1)| Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example (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. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example (continued) b) f (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. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example (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. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 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 + 0.05. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example a. The population of Savannah, Georgia, is 132,410. Find the average walking speed of people living in Savannah. The population of Philadelphia, Pennsylvania, is 1,540,351. Find the average walking speed of people living in Philadelphia. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example (continued) Solution: a. Since P is in thousands and 132,410 = 132.410 thousand, we substitute 132.410 for P: w(132.410) = 0.37 ln 132.410 + 0.05  1.9 ft/sec. The average walking speed of people living in Savannah is about 1.9 ft/sec. b. Substitute 1540.351 for P: w(1540.351) = 0.37 ln 1540.351 + 0.05  2.8 ft/sec. The average walking speed of people living in Philadelphia is about 2.8 ft/sec. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley