Mathematics
Integration of Logarithmic Functions
The integration of logarithmic functions involves finding the antiderivative of functions involving logarithms. This process often requires using techniques such as substitution or integration by parts. The resulting integral may involve logarithmic terms and can be used to solve various mathematical problems, particularly in calculus and mathematical modeling.
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9 Key excerpts on "Integration of Logarithmic Functions"
eBook - PDF- Tom M. Apostol(Author)
- 2019(Publication Date)
- Wiley(Publisher)
6.2 Motivation for the definition of the natural logarithm as an integral The logarithm is an example of a mathematical concept that can be defined in many different ways. When a mathematician tries to formulate a definition of a concept, such as the logarithm, he usually has in mind a number of properties he wants this concept to have. By examining these properties, he is often led to a simple formula or process that might serve as a definition from which all the desired properties spring forth as logical deductions. We shall illustrate how this procedure may be used to arrive at the definition of the logarithm which is given in the next section. One of the properties we want logarithms to have is that the logarithm of a product should be the sum of the logarithms of the individual factors. Let us consider this property by itself and see where it leads us. If we think of the logarithm as a function f , then we want this function to have the property expressed by the formula f (xy) = f (x) + f (y) (6.4) whenever x, y, and xy are in the domain of f . An equation like (6.4), which expresses a relationship between the values of a function at two or more points, is called a functional equation. Many mathematical problems can be reduced to solving a functional equation, a solution being any function which satisfies the equation. Ordinarily an equation of this sort has many different solutions, and it is usually very difficult to find them all. It is easier to seek only those solutions which have some additional property such as continuity or differentiability. For the most part, these are the only solutions we are interested in anyway. We shall adopt this point of view and determine all differentiable solutions of (6.4). But first let us try to deduce what information we can from (6.4) alone, without any further restrictions on f . One solution of (6.4) is the function that is zero everywhere on the real axis.
eBook - PDFCalculus
Single Variable
- Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
- 2022(Publication Date)
- Wiley(Publisher)
6.2 Derivatives and Integrals Involving Logarithmic Functions 345 Logarithmic Differentiation We now consider a technique called logarithmic differentiation that is useful for differentiating functions that are composed of products, quotients, and powers. Example 5 The derivative of y = x 2 3 √ 7x − 14 (1 + x 2 ) 4 (7) is messy to calculate directly. However, if we first take the natural logarithm of both sides and then use its properties, we can write ln y = 2 ln x + 1 3 ln(7x − 14) − 4ln(1 + x 2 ) Differentiating both sides with respect to x yields 1 y dy dx = 2 x + 7 / 3 7x − 14 − 8x 1 + x 2 Thus, on solving for dy / dx and using (7) we obtain dy dx = x 2 3 √ 7x − 14 (1 + x 2 ) 4 2 x + 1 3x − 6 − 8x 1 + x 2 Remark Since ln y is only defined for y > 0, the computations in Example 5 are only valid for x > 2 (verify). However, using derivative rules it can be shown that the formula obtained for dy/dx is valid for x < 2, x = 0, as well as x > 2. In general, whenever a derivative dy/dx is obtained by logarithmic differentiation, the resulting derivative formula will be valid for all values of x for which y = 0. It may be valid at those points as well, but it is not guaranteed (Exercises 76–78). Integrals Involving ln x Formula (2) states that the function ln x is an antiderivative of 1 / x on the interval (0, +∞), whereas Formula (6) states that the function ln |x| is an antiderivative of 1 / x on each of the intervals (−∞, 0) and (0, +∞). Thus we have the companion integration formula to (6), 1 u du = ln |u| + C (8) with the implicit understanding that this formula is applicable only across an interval that does not contain 0. Example 6 Applying Formula (8), we find e 1 1 x dx = ln |x| e 1 = ln |e| − ln |1| = 1 − 0 = 1 −1 −e 1 x dx = ln |x| −1 −e = ln | −1| − ln | −e| = 0 − 1 = −1 Example 7 Evaluate 3x 2 x 3 + 5 dx.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Orange Apple(Publisher)
Moreover, analytic properties of the function pass to its inverse function. Thus, as the exponential function f ( x ) = b x is continuous and differentiable, so is its inverse function, log b ( x ). Roughly speaking, a differentiable function is one whose graph has no sharp corners. Derivative and antiderivative The area of the hyperbolic sector equals ln( b ) − ln( a ). The derivative of the natural logarithm ln( x ) = log e ( x ) is given by This can be derived from the definition as the inverse function of e x , using the chain rule. This implies that the antiderivative of 1/ x is ln( x ) + C. An early application of this fact was the quadrature of a hyperbolic sector, as shown at the right, by de Saint-Vincent in 1647. The derivative with a generalised functional argument f ( x ) is ________________________ WORLD TECHNOLOGIES ________________________ For this reason the quotient at right hand side is called logarithmic derivative of f . The antiderivative of the natural logarithm ln( x ) is Derivatives and antiderivatives of logarithms to other bases can be derived therefrom using the formula for change of bases. Integral representation of the natural logarithm The natural logarithm of t is the shaded area underneath the graph of the function f ( x ) = 1/ x (reciprocal of x ). The natural logarithm satisfies the following identity: In prose, the natural logarithm of t agrees with the integral of 1/ x dx from 1 to t , that is to say, the area between the x -axis and the function 1/ x , ranging from x = 1 to x = t . This is depicted at the right. The formula is a consequence of the fundamental theorem of calculus and the above formula for the derivative of ln( x ). Some authors actually use the right hand side of this equation as a definition of the natural logarithm and derive the formulas concerning logarithms of products and powers mentioned above from this definition. The product formula ln( tu ) = ln( t ) + ln( u ) is deduced in the following way:
- Gautam Bandyopadhyay(Author)
- 2023(Publication Date)
- CRC Press(Publisher)
Main impetus in this regard came from astronomy where it was frequently necessary to multiply and divide large numbers. However, logarithm can be perceived from many other angles. It can be viewed as the area under the rectangular hyperbola y = 1 x in geometry. It can be used as the inverse of exponential function e x or a x. As such we may treat it as the inverse of continuous compounding problem when we are interested to know in how many years Rs. 1/- will have a matured value e x or a x. In analysis we find that it is the limit of the product of two factors which are functions of n when n tends to infinity. It can also be expressed as an infinite series. It is one of the core functions in mathematics extended to negative and complex numbers. It plays vital roles in many branches of mathematics. Mathematical expressions for inductance and capacitance of a transmission line contain logarithmic terms. Logarithm forms the basis of Richter scale and measure of pH. It has wide applications in many other fields as well. 3.2 Logarithm as artificial numbers facilitating computation “Logarithms are a set of artificial numbers invented and formed into tables for the purpose of facilitating arithmetical computations. They are adapted to the natural numbers in such a manner that by their aid Addition supplies the place of Multiplication, Subtraction to that of Division, Multiplication that of Involution, and Division that of Evolution or the Extraction of Roots”. Excerpt from A Manual of Logarithms and Practical Mathematics for the use of students, Engineers, Navigators and Surveyors — by James Trotter of Edinburgh Published by Oliver & Boyd, Tweeddale Court and Simpkin, Marshall, & Co. London in 1841. In eleventh century Ibon Jonuis, an Arab mathematician proposed a method of multiplication which can save computational labour significantly. The method is known as Prosthaphaeresis. The Greek word prosthesis means addition and aphaeresis means subtraction
- Ronald Harshbarger, James J. Reynolds(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
This requires the use of logarithmic functions. In this section, we will evaluate and graph logarithmic functions, find logarithmic functions that model data, and use properties of logarithmic functions. SEC TION 5.2 OBJECTIVES • To use and apply the definition of logarithmic functions • To graph logarithmic functions • To model logarithmic functions • To use properties of logarithmic functions • To use the change-of- base formula Copyright 2019 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. 330 CHAPTER 5 Exponential and Logarithmic Functions Logarithmic Functions and Graphs Before the development and easy availability of calculators and computers, certain arith- metic computations, such as (1.37) 13 and !3.09 , 16 were difficult to perform. The computa- tions could be performed relatively easily using logarithms, which were developed in the seventeenth century by John Napier, or using a slide rule, which is based on logarithms. The use of logarithms as a computing technique has all but disappeared today, but the study of logarithmic functions is still very important because of the many applications of these functions. For example, let us again consider the culture of bacteria described at the beginning of the previous section.
eBook - PDFCalculus
Single Variable
- Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
- 2016(Publication Date)
- Wiley(Publisher)
350 Chapter 6 / Exponential, Logarithmic, and Inverse Trigonometric Functions LOGARITHMIC DIFFERENTIATION We now consider a technique called logarithmic differentiation that is useful for differen- tiating functions that are composed of products, quotients, and powers. Example 5 The derivative of y = x 2 3 √ 7x − 14 (1 + x 2 ) 4 (7) is messy to calculate directly. However, if we first take the natural logarithm of both sides and then use its properties, we can write ln y = 2 ln x + 1 3 ln(7x − 14) − 4ln(1 + x 2 ) Differentiating both sides with respect to x yields 1 y dy dx = 2 x + 7 / 3 7x − 14 − 8x 1 + x 2 Thus, on solving for dy /dx and using (7) we obtain dy dx = x 2 3 √ 7x − 14 (1 + x 2 ) 4 2 x + 1 3x − 6 − 8x 1 + x 2 REMARK Since ln y is only defined for y > 0, the computations in Example 5 are only valid for x > 2 (verify). However, using the fact that the expression for the derivative of ln y is the same as that for ln |y|, it can be shown that the formula obtained for dy / dx is valid for x < 2 as well as x > 2 (Exercise 75). In general, whenever a derivative dy / dx is obtained by logarithmic differentiation, the resulting derivative formula will be valid for all values of x for which y = 0. It may be valid at those points as well, but it is not guaranteed. INTEGRALS INVOLVING ln x Formula (2) states that the function ln x is an antiderivative of 1 / x on the interval (0, +∞), whereas Formula (6) states that the function ln |x| is an antiderivative of 1 / x on each of the intervals (−∞, 0) and (0, +∞). Thus we have the companion integration formula to (6), 1 u du = ln |u| + C (8) with the implicit understanding that this formula is applicable only across an interval that does not contain 0. Example 6 Applying Formula (8), e 1 1 x dx = ln |x|] e 1 = ln |e| − ln |1| = 1 − 0 = 1 −1 −e 1 x dx = ln |x|] −1 −e = ln | −1| − ln | − e| = 0 − 1 = −1 Example 7 Evaluate 3x 2 x 3 + 5 dx.
eBook - PDF- David Cohen, Theodore Lee, David Sklar, , David Cohen, Theodore Lee, David Sklar(Authors)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
5.3 Logarithmic Functions 351 EXAMPLE 4 Finding the Domain of a Function Defined by a Logarithm Find the domain of the function f ( x ) log 2 (12 4 x ). SOLUTION As you can see by looking back at Figure 3 on page 348, the inputs for the log-arithmic function must be positive. So, in the case at hand, we require that the quan-tity 12 4 x be positive. Consequently, we have Therefore the domain of the function f ( x ) log 2 (12 4 x ) is the interval ( q , 3). The next example concerns the exponential function y e x and its inverse func-tion, y log e x . Many books, as well as calculators, abbreviate the expression log e x by ln x , read natural log of x .* For reference and emphasis we repeat this fact in the following box. (Incidentally, on most calculators, “log” is an abbreviation for log 10 .) x 3 4 x 12 12 4 x 0 *According to the historian Florian Cajori, the notation ln x was used by (and perhaps first introduced by) Irving Stringham in his text Uniplanar Algebra (San Francisco: University Press, 1893). Definition The “ln” Notation for Base e Logarithms ln x means log e x EXAMPLE 1. ln e 1 because ln e stands for log e e , which equals 1. 2. ln( e 2 ) 2 because ln( e 2 ) stands for log e ( e 2 ), which equals 2. 3. ln 1 0 because ln 1 stands for log e 1, which equals 0. (The exponential form of the equation ln 1 0 is e 0 1.) Copyright 201 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. 352 CHAPTER 5 Exponential and Logarithmic Functions EXAMPLE 5 Sketching the Graph of ln x and a Translation Graph the following functions: (a) y ln x ; (b) y ln ( x 1) 1.
eBook - PDF- Cynthia Y. Young(Author)
- 2021(Publication Date)
- Wiley(Publisher)
5.2 Logarithmic Functions and Their Graphs 447 All of the transformation techniques (shifting, reflection, and compression) discussed in Chapter 3 also apply to logarithmic functions. For example, the graphs of −log 2 x and log 2 (−x) are found by reflecting the graph of y = log 2 x about the x-axis and y-axis, respectively. x (2, 1) (2, –1) (– 4, 2) (4, 2) (4, –2) (–2, 1) Concept Check Find the x-intercept, domain, and range of log b (x − 1). Answer: x-intercept: (2, 0); Domain: (1, ∞); Range: (−∞, ∞). Video EXAMPLE 8 Graphing Logarithmic Functions Using Transformations Graph the function f (x) = −log 2 (x − 3) and state its domain and range. Solution Graph y = log 2 x. x-intercept: (1, 0) Vertical asymptote: x = 0 Additional points: (2, 1), (4, 2) Graph y = log 2 (x − 3) by shifting y = log 2 x to the right three units. x-intercept: (4, 0) Vertical asymptote: x = 3 Additional points: (5, 1), (7, 2) Graph y = −log 2 (x − 3) by reflecting y = log 2 (x − 3) about the x-axis. x-intercept: (4, 0) Vertical asymptote: x = 3 Additional points: (5, −1), (7, −2) Domain: (3, ∞) Range: (−∞, ∞) x y 10 5 –5 (4, 2) (2, 1) (1, 0) x y 10 5 –5 (7, 2) (5, 1) (4, 0) x y 10 5 –5 (4, 0) (5, –1) (7, –2) 448 CHAPTER 5 Exponential and Logarithmic Functions 5.2.4 Applications of Logarithms 5.2.4 Skill Apply logarithmic functions to problems in the natural sciences and engineering. 5.2.4 Conceptual Understand that logarithmic functions allow very large ranges of numbers in science and engineering applications to be represented on a smaller scale. Logarithms are used to make a large range of numbers manageable. For example, to create a scale to measure a human’s ability to hear, we must have a way to measure the sound intensity of an explosion, even though that intensity can be more than a trillion (10 12 ) times greater than that of a soft whisper. Decibels in engineering and physics, pH in chemistry, and the Richter scale for earthquakes are all applications of logarithmic functions.
eBook - PDFIntermediate Algebra
Connecting Concepts through Applications
- Mark Clark, Cynthia Anfinson(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
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. C H A P T E R 6 L o g a r i t h m i c F u n c t i o n s 574 To plot the logarithm graph, we need a large scale for the x-axis and a small scale for the y-axis. Note that this graph does not touch or cross the y-axis. The y-axis is a vertical asymptote. –3 –6 –12 –9 y x –250 3 750 1250 250 0 PRACTICE PROBLEM FOR EXAMPLE 2 Sketch the graph of f 1 x 2 5 log 0.25 x. Label and scale the axes. We see from both of these examples that basic logarithm graphs have a few things in common. The y-axis is a vertical asymptote, the x-intercept is 1 1, 0 2 , and the graphs grow or decay rapidly at first and then slow down as the inputs increase. Domain and Range of Logarithmic Functions There are several ways to think about the domain and range of a logarithmic function. The inverse relationship between logarithmic and exponential functions can be used to relate the domain and range of exponential functions to the domain and range of logarithmic functions. We can also consider the graphs of logarithmic functions to find their domain and range. In Chapter 5, we found that the domain and range of an exponential function of the form f 1 x 2 5 a # b x were all real numbers, 1 2`, `2 , and all positive real numbers, 1 0, `2 , respectively. Since logarithms are the inverse of exponentials, the domain and range are switched. Therefore, the domain for a logarithm function of the form f 1 x 2 5 log b x will be all positive real numbers or 1 0, `2 , and the range will be all real numbers or 1 2`, ` 2 .
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