318 Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions 5.1 The Natural Logarithmic Function: Differentiation Develop and use properties of the natural logarithmic function. Understand the definition of the number e. Find derivatives of functions involving the natural logarithmic function. The Natural Logarithmic Function Recall that the General Power Rule x nϩ1 ͵x n dx ϭ ϩ C, n Ϫ1 General Power Rule n ϩ 1 has an important disclaimer—it doesn’t apply when n ϭϪ1. Consequently, you have not yet found an antiderivative for the function f ͑x͒ ϭ 1͞x. In this section, you will use the Second Fundamental Theorem of Calculus to define such a function. This antiderivative is a function that you have not encountered previously in the text. It is neither algebraic nor trigonometric, but falls into a new class of functions called logarithmic functions. This particular function is the natural logarithmic function. JOHN NAPIER (1550–1617) Definition of the Natural Logarithmic Function Logarithms were invented by the Scottish mathematician John The natural logarithmic function is defined by Napier. Napier coined the term x 1 logarithm, from the two Greek ln x ϭ ͵ dt, x > 0. words logos (or ratio) and 1 t arithmos (or number), to describe the theory that he spent 20 years The domain of the natural logarithmic function is the set of all positive real developing and that first appeared numbers. in the book Mirifici Logarithmorum canonis descriptio (A Description of the Marvelous Rule of Logarithms). Although he did not From this definition, you can see that ln x is positive for x > 1 and negative for introduce the natural logarithmic 0 < x < 1, as shown in Figure 5.1. Moreover,ln͑1͒ ϭ 0, because the upper and lower function, it is sometimes called the limits of integration are equal when x ϭ 1. Napierian logarithm. See LarsonCalculus.com to read y y more of this biography. 4 4 y = 1 y = 1 t t 3 3 x x If x > 1, then ∫ 1 dt > 0. If 0 < x < 1, then ∫ 1 dt < 0. 2 1 t 2 1 t 1 1 x t x t 1 2 3 4 1 2 3 4 If x > 1, then ln x > 0. If 0 < x < 1, then ln x < 0. Figure 5.1 Exploration Graphing the Natural Logarithmic Function Using only the definition of the natural logarithmic function, sketch a graph of the function. Explain your reasoning. Mary Evans Picture Library Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 5.1 The Natural Logarithmic Function: Differentiation 319 y To sketch the graph of y ϭ ln x, you can think of the natural logarithmic function y = ln x as an antiderivative given by the differential equation 1 (1, 0) dy 1 x ϭ . 1 2345 dx x − 1 Figure 5.2 is a computer-generated graph, called a slope field (or direction field), ͞ ϭ −2 showing small line segments of slope 1 x. The graph of y ln x is the solution that passes through the point ͑1, 0͒. (You will study slope fields in Section 6.1.) −3 The next theorem lists some basic properties of the natural logarithmic function. 1 Each small line segment has a slope of . x THEOREM 5.1 Properties of the Natural Logarithmic Function Figure 5.2 The natural logarithmic function has the following properties. 1. The domain is ͑0, ϱ͒ and the range is ͑Ϫϱ, ϱ͒. 2. The function is continuous, increasing, and one-to-one. 3. The graph is concave downward. y y′ = 1 Proof The domain of f ͑x͒ ϭ ln x is ͑0, ϱ͒ by definition. Moreover, the function is y′ = 1 4 3 continuous because it is differentiable. It is increasing because its derivative y′ = 1 x = 4 1 2 x = 3 y = ln x 1 y′ = 1 x = 2 fЈ ͑x͒ ϭ First derivative x x y′ = 2 1 234 x = 1 is positive for x > 0, as shown in Figure 5.3. It is concave downward because −1 x = 1 y′ = 3 2 1 x = 1 fЉ ͑x͒ ϭϪ Second derivative y′ = 4 3 x2 −2 x = 1 4 is negative for x > 0. The proof that f is one-to-one is given in Appendix A. The The natural logarithmic function is increasing, and its graph is concave following limits imply that its range is the entire real number line. downward. ϭϪϱ limϩ ln x Figure 5.3 x→0 and lim ln x ϭ ϱ x→ϱ Verification of these two limits is given in Appendix A. See LarsonCalculus.com for Bruce Edwards’s video of this proof. Using the definition of the natural logarithmic function, you can prove several important properties involving operations with natural logarithms. If you are already familiar with logarithms, you will recognize that these properties are characteristic of all logarithms. THEOREM 5.2 Logarithmic Properties If a and b are positive numbers and n is rational, then the following properties are true. 1. ln͑1͒ ϭ 0 2. ln͑ab͒ ϭ ln a ϩ ln b 3. ln͑an͒ ϭ n ln a a 4. ln΂ ΃ ϭ ln a Ϫ ln b b Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 320 Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions Proof The first property has already been discussed. The proof of the second property follows from the fact that two antiderivatives of the same function differ at most by a constant. From the Second Fundamental Theorem of Calculus and the definition of the natural logarithmic function, you know that d d x 1 1 ͓ln x͔ ϭ ΄͵ dt΅ ϭ . dx dx 1 t x So, consider the two derivatives d a 1 ͓ln͑ax͔͒ ϭ ϭ dx ax x and d 1 1 ͓ln a ϩ ln x͔ ϭ 0 ϩ ϭ . dx x x Because ln͑ax͒ and ͑ln a ϩ ln x͒ are both antiderivatives of 1͞x, they must differ at most by a constant. ln͑ax͒ ϭ ln a ϩ ln x ϩ C By letting x ϭ 1, you can see that C ϭ 0. The third property can be proved similarly by comparing the derivatives of ln͑xn͒ and n lnx. Finally, using the second and third properties, you can prove the fourth property. a ln΂ ΃ ϭ ln͓a͑bϪ1͔͒ ϭ ln a ϩ ln͑bϪ1͒ ϭ ln a Ϫ ln b b See LarsonCalculus.com for Bruce Edwards’s video of this proof. Example 1 shows how logarithmic properties can be used to expand logarithmic expressions. Expanding Logarithmic Expressions 10 a. ln ϭ ln 10 Ϫ ln 9 Property 4 9 f(x) = ln x2 ͞ 5 b. lnΊ3x ϩ 2 ϭ ln͑3x ϩ 2͒1 2 Rewrite with rational exponent. 1 ϭ ln͑3x ϩ 2͒ Property 3 2 −5 5 6x c. ln ϭ ln͑6x͒ Ϫ ln 5 Property 4 5 ϭ ln 6 ϩ ln x Ϫ ln 5 Property 2 2 2 −5 ͑x ϩ 3͒ d. ln ϭ ln͑x 2 ϩ 3͒ 2 Ϫ ln͑xΊ3 x 2 ϩ 1 ͒ xΊ3 x 2 ϩ 1 ϭ ͑ 2 ϩ ͒ Ϫ ͓ ϩ ͑ 2 ϩ ͒1͞3͔ g(x) = 2 ln x 2 ln x 3 ln x ln x 1 5 ϭ 2 ln͑x 2 ϩ 3͒ Ϫ ln x Ϫ ln͑x 2 ϩ 1͒1͞3 1 ϭ 2 ln͑x 2 ϩ 3͒ Ϫ ln x Ϫ ln͑x 2 ϩ 1͒ 3 −5 5 When using the properties of logarithms to rewrite logarithmic functions, you must check to see whether the domain of the rewritten function is the same as the domain of −5 the original. For instance, the domain of f͑x͒ ϭ ln x 2 is all real numbers except x ϭ 0, Figure 5.4 and the domain of g͑x͒ ϭ 2 ln x is all positive real numbers. (See Figure 5.4.) Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 5.1 The Natural Logarithmic Function: Differentiation 321 The Number e It is likely that you have studied logarithms in an y algebra course. There, without the benefit of calculus, 3 logarithms would have been defined in terms of a y = 1 t base number. For example, common logarithms have a base of 10 and therefore log 10 ϭ 1. (You 2 10 e Area = ∫ 1 dt = 1 will learn more about this in Section 5.5.) 1 t The base for the natural logarithm is defined 1 using the fact that the natural logarithmic function is continuous, is one-to-one, and has a range of t ͑Ϫϱ, ϱ͒.