Logarithmic Functions

12 Key excerpts on "Logarithmic Functions"

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  Precalculus

  Functions and Graphs

  • Earl Swokowski, Jeffery Cole(Authors)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  249 4 4.1 Inverse Functions 4.2 Exponential Functions 4.3 The Natural Exponential Function 4.4 Logarithmic Functions 4.5 Properties of Logarithms 4.6 Exponential and Logarithmic Equations EXPONENTIAL AND Logarithmic Functions are transcendental functions, since they cannot be defined in terms of only addition, sub-traction, multiplication, division, and rational powers of a variable x , as is the case for the algebraic functions considered in previous chapters. Such functions are of major importance in mathematics and have applications in almost every field of human endeavor. They are especially useful in the fields of chemistry, biology, physics, and engineering, where they help describe the manner in which quantities in nature grow or decay. As we shall see in this chapter, there is a close relationship between specific exponential and Logarithmic Functions—they are inverse functions of each other. Inverse, Exponential, and Logarithmic Functions 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. A function f may have the same value for different numbers in its domain. For example, if f s x d 5 x 2 , then f s 2 d 5 4 and f s 2 2 d 5 4 , but 2 ± 2 2 . For the inverse of a function to be defined, it is essential that different numbers in the domain always give different values of f . Such functions are called one-to-one functions. The arrow diagram in Figure 1 illustrates a one-to-one function. Note that each function value in the range R corresponds to exactly one element in the domain D .

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  Calculus, Volume 1

  • Tom M. Apostol(Author)
  • 2019(Publication Date)
  • Wiley

    (Publisher)

  6 THE LOGARITHM, THE EXPONENTIAL, AND THE INVERSE TRIGONOMETRIC FUNCTIONS 6.1 Introduction Whenever man focuses his attention on quantitative relationships, he is either studying the properties of a known function or trying to discover the properties of an unknown function. The function concept is so broad and so general that it is not surprising to find an endless variety of functions occurring in nature. What is surprising is that a few rather special functions govern so many totally different kinds of natural phenomena. We shall study some of these functions in this chapter—first of all, the logarithm and its inverse (the exponential function) and secondly, the inverses of the trigonometric functions. Anyone who studies mathematics, either as an abstract discipline or as a tool for some other scientific field, will find that a good working knowledge of these functions and their properties is indispensable. The reader probably has had occasion to work with logarithms to the base 10 in an elemen- tary algebra or trigonometry course. The definition usually given in elementary algebra is this: If x > 0, the logarithm of x to the base 10, denoted by log 10 x, is that real number u such that 10 u = x. If x = 10 u and y = 10 v , the law of exponents yields xy = 10 u+v . In terms of logarithms, this becomes log 10 (xy) = log 10 x + log 10 y. (6.1) It is this fundamental property that makes logarithms particularly adaptable to computations involving multiplication. The number 10 is useful as a base because real numbers are commonly written in the decimal system, and certain important numbers like 0.01, 0.1, 1, 10, 100, 1000, . . . have for their logarithms the integers −2, −1, 0, 1, 2, 3, . . . , respectively. It is not necessary to restrict ourselves to base 10. Any other positive base b ≠ 1 would serve equally well. Thus u = log b x means x = b u , (6.2) and the fundamental property in (6.1) becomes log b (xy) = log b x + log b y. (6.3) 226

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  College Algebra

  • James Stewart, Lothar Redlin, Saleem Watson, , James Stewart, Lothar Redlin, Saleem Watson(Authors)
  • 2015(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  In this chapter we study exponential functions . These are functions like f 1 x 2  2 x , where the independent variable is in the exponent. Exponential functions are used in modeling many real-world phenomena, such as the growth of a population, the growth of an investment that earns compound interest, or the decay of a radioactive substance. Once an exponential model has been obtained, we can use the model to predict the size of a population, calculate the amount of an investment, or find the amount of a radioactive substance that remains. The inverse functions of exponential functions are called Logarithmic Functions . With exponential models and Logarithmic Functions we can answer questions such as these: When will my city be as crowded as the city street pictured here? When will my bank account have a million dollars? When will radiation from a radioactive spill decay to a safe level? In the Focus on Modeling at the end of the chapter we learn how to fit exponential and power curves to data. 365 Exponential and Logarithmic Functions 4 4.1 Exponential Functions 4.2 The Natural Exponential Function 4.3 Logarithmic Functions 4.4 Laws of Logarithms 4.5 Exponential and Logarithmic Equations 4.6 Modeling with Exponential Functions 4.7 Logarithmic Scales FOCUS ON MODELING Fitting Exponential and Power Curves to Data © TonyV3112/Shutterstock.com Copyright 2016 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.

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  Intermediate Algebra

  Concepts and Graphs 2E

  • Charles P. McKeague(Author)
  • 2019(Publication Date)
  • XYZ Textbooks

    (Publisher)

  674 CHAPTER 8 Exponential and Logarithmic Functions Now, this last equation is actually the equation of a logarithmic function, as the following definition indicates: Notation When an expression is in the form x = b y , it is said to be in exponential form. On the other hand, if an expression is in the form y = log b x, it is said to be in logarithmic form. Here are some equivalent statements written in both forms. Logarithmic Equations EXAMPLE 1 Solve for x: log 3 x = −2. SOLUTION In exponential form, the equation looks like this: x = 3 −2 or x = 1 __ 9 The solution is 1 _ 9 . EXAMPLE 2 Solve log x 4 = 3. SOLUTION Again, we use the definition of logarithms to write the expression in exponential form: 4 = x 3 Taking the cube root of both sides, we have 3 √ — 4 = 3 √ — x 3 x = 3 √ — 4 The solution set is { 3 √ — 4}. DEFINITION The expression y = log b x is read “y is the logarithm to the base b of x” and is equivalent to the expression x = b y b > 0, b ≠ 1 In words, we say “y is the number we raise b to in order to get x.” Exponential Form Logarithmic Form 8 = 2 3 log 2 8 = 3 25 = 5 2 log 5 25 = 2 0.1 = 10 −1 log 10 0.1 = −1 1 _ 8 = 2 −3 log 2 1 _ 8 = −3 r = z s log z r = s b 8.3 Logarithms Are Exponents 675 EXAMPLE 3 Solve log 8 4 = x. SOLUTION We write the expression again in exponential form: 4 = 8 x Because both 4 and 8 can be written as powers of 2, we write them in terms of powers of 2: 2 2 = (2 3 ) x 2 2 = 2 3x The only way the left and right sides of this last line can be equal is if the exponents are equal — that is, if 2 = 3x or x = 2 __ 3 The solution is 2 _ 3 . We check as follows: log 8 4 = 2 __ 3 4 = 8 2/3 4 = ( 3 √ — 8) 2 4 = 2 2 4 = 4 The solution checks when used in the original equation. Graphing Logarithmic Functions Graphing Logarithmic Functions can be done using the graphs of exponential func- tions and the fact that the graphs of inverse functions have symmetry about the line y = x.

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  Algebra and Trigonometry

  • Cynthia Y. Young(Author)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  In light of this, comment on the shortcomings of the best fit exponential curve. 440 CHAPTER 5 Exponential and Logarithmic Functions 5.2 Logarithmic Functions and Their Graphs SKILLS OBJECTIVES • Evaluate logarithmic expressions. • Approximate common and natural logarithms using a calculator. • Graph Logarithmic Functions. • Apply Logarithmic Functions to problems in the natural sciences and engineering. CONCEPTUAL OBJECTIVES • Interpret Logarithmic Functions as inverses of exponential functions. • Recognize when a logarithm or the value of a logarithmic function can be evaluated exactly or when it must be approximated. • Understand the inverse relationship between the characteristics of Logarithmic Functions and exponential functions. • Understand that Logarithmic Functions allow very large ranges of numbers in science and engineering applications to be represented on a smaller scale. 5.2.1 Evaluating Logarithms 5.2.1 Skill Evaluate logarithmic expressions. 5.2.1 Conceptual Interpret Logarithmic Functions as inverses of exponential functions. In Section 5.1, we found that the graph of an exponential function f (x) = b x passes through the point (0, 1), with the x-axis as a horizontal asymptote. The graph passes both the vertical line test (for a function) and the horizontal line test (for a one-to-one function), and therefore an inverse exists. We will now apply the technique outlined in Section 3.5 to find the inverse of f (x) = b x : Words Math Let y = f (x). y = b x Interchange x and y. x = b y Solve for y. y = ? We see that y is the exponent that b is raised to in order to obtain x. We call this exponent a logarithm (or “log” for short). Words Math x = b y is equivalent to y = log b x. y = log b x Let y = f −1 (x). f −1 (x) = log b x Logarithmic Function For x > 0, b > 0, and b ≠ 1, the logarithmic function with base b is denoted f (x) = log b x, where y = log b x if and only if x = b y We read log b x as “log base b of x.”

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  The R Student Companion

  • Brian Dennis(Author)
  • 2016(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  Because y x = e means x y = ( ) l og , we can think of taking logarithms of both sides of y x = e in order to bring down x : y x = e , l og l og y x x ( ) = ( ) = . e The exponential and Logarithmic Functions are inverse functions of each other. Inverse here does not mean one is the reciprocal of the other. Rather, it means one erases the other’s doings: l og e x x ( ) = , e l og y y ( ) = , 175 Exponential and Logarithmic Functions where x is a real number and y is a positive real number. Raising e to a power x is sometimes called “taking the antilogarithm of x ” and is accessed on some scientific calculators as “ inv-log ” or “ inv-ln .” Also, if a is a positive real number, then a a = ( ) e l og and, so, taking the loga-rithm of a x brings down x multiplied by l o g a ( ) : a x a x = ( ) [ ] e l og , l og l og . a a x x ( ) = ( ) [ ] The above formula is the key to going back and forth between logarithms in base e and other bases. If y a a x y a x a = = = ( ) ( ) [ ] , l og l og e then l og l og l og y a y a ( ) = ( ) [ ] ( ) ⎡ ⎣ ⎤ ⎦ , and so the log to any base a expressed in terms of base e logarithms is l og l og l og a y y a ( ) = ( ) ⎡ ⎣ ⎤ ⎦ ( ) [ ] / . In R, a function for l og a y ( ) is log(y,a). Logarithmic Scales In science, some phenomena are measured for convenience on a logarithmic scale. A logarithmic scale might be used for a quantity that has an enormous range of values or that varies multiplicatively. Richter Scale A well-known example of a logarithmic scale is the Richter scale for measur-ing the magnitude of earthquakes. The word magnitude gives a clue that the scale is logarithmic. Richter magnitude is defined as the base 10 logarithm of the amplitude of the quake waves recorded by a seismograph (amplitude is the distance of departures of the seismograph needle from its central refer-ence point). Each whole number increase in magnitude represents a quake with waves measuring 10 times greater.

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  College Algebra

  Building Skills and Modeling Situations

  • Charles P. McKeague, Katherine Yoshiwara, Denny Burzynski(Authors)
  • 2013(Publication Date)
  • XYZ Textbooks

    (Publisher)

  Chapter Outline 6.1 Exponential Functions 6.2 The Inverse of a Function 6.3 Logarithms Are Exponents 6.4 Properties of Logarithms 6.5 Common Logarithms and Natural Logarithms 6.6 Exponential Equations and Change of Base Study Skills 414 Never mistake activity for achievement. — John Wooden, legendary UCLA basketball coach You may think that this John Wooden quote has to do with being productive and efficient, or using your time wisely, but it is really about being honest with your- self. I have had students come to me after failing a test saying, “I can’t understand why I got such a low grade after I put so much time in studying.” One student even had help from a tutor and felt she understood everything that we covered. After asking her a few questions, it became clear that she spent all her time studying with a tutor and the tutor was doing most of the work. The tutor can work all the home- work problems, but the student cannot. She has mistaken activity for achievement. Can you think of situations in your life when you are mistaking activity for achievement? How would you describe someone who is mistaking activity for achievement in the way they study for their math class? Which of the following best describes the idea behind the John Wooden quote? ▶ Always be efficient. ▶ Don’t kid yourself. ▶ Take responsibility for your own success. ▶ Study with purpose. 6.1 Exponential Functions 415 6.1 INTRODUCTION To obtain an intuitive idea of how exponential functions behave, we can consider the heights attained by a bouncing ball. When a ball used in the game of racquetball is dropped from any height, the first bounce will reach a height that is 2 _ 3 of the original height. The second bounce will reach 2 _ 3 of the height of the first bounce, and so on, as shown in Figure 1. If the ball is initially dropped from a height of 1 meter, then during the first bounce it will reach a height of 2 _ 3 meter.

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  Precalculus, Enhanced Edition

  • 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.

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  Intermediate Algebra

  Connecting Concepts through Applications

  • Mark Clark, Cynthia Anfinson(Authors)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Find the domain and range of Logarithmic Functions. 6.3 Graphing Logarithmic Functions In Section 6.2, we learned that Logarithmic Functions are defined as inverses of exponential functions. This inverse relationship helps us to investigate the graph of logarithm functions using the information we know about the related exponential graphs. Recall from Section 6.1 that the graph of an inverse function is a reflection of the graph of the original function over the line y 5 x. With this relationship in mind, let’s look at some exponential graphs and their related logarithm graphs. 1. Fill in the following table of values for the function f 1 x 2 5 2 x . x 22 21 0 1 2 3 4 5 6 7 8 f 1 x 2 5 2 x 2 22 5 1 4 2. What kind of scale should we use on the x-axis for this graph? 0.5 1 2 5 10 other: _________ 3. What kind of scale should we use on the y-axis for this graph? 0.5 1 2 5 10 other: _________ 4. Sketch the graph of the function in part 1. 5. Use the table from part 1 to create a table for the inverse function. (Remember that the x- and y-values will be swapped.) x 1 4 f 21 1 x 2 5 log 2 x 5 log x log 2 22 How do we build a logarithm graph? CONCEPT INVESTIGATION Skill Connection Scale Remember that the scale is the spacing on the axes. The spacing on each axis must be consistent so that each space between tick marks represents the same number of units. 105. log 16 m 5 1 2 106. log 8 x 5 1 3 107. log 12 x 5 1 5 108. log 11 x 5 2 3 109. log 1 8t 2 5 2 110. log 1 2x 2 5 0.5 111. ln 1 3x 2 5 4 112. log 7 1 4x 2 5 3 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.

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  College Algebra

  Building Concepts and Connections 2E

  • Revathi Narasimhan(Author)
  • 2019(Publication Date)
  • XYZ Textbooks

    (Publisher)

  Chapter 4 T he value of a new car depreciates over time. This phenomenon can be modeled by an exponential function, a type of function in which the independent variable appears in the exponent. A simple illustration of this model is given in Example 9 in Section 4.2. This chapter will explore exponential functions, and their inverse, Logarithmic Functions. These functions are invaluable in the study of more advanced mathematics and have numerous applications in a variety of fields, including engineering, the life sciences, business, physics and computer science. Outline 4.1 Inverse Functions 4.2 Exponential Functions 4.3 Logarithmic Functions 4.4 Properties of Logarithms 4.5 Exponential and Logarithmic Equations 4.6 Exponential, Logistic, and Logarithmic Models Exponential and Logarithmic Functions 4 . 1 320 Chapter 4 Exponential and Logarithmic Functions Inverse Functions In Section 2.7, we discussed the composition of functions, which involves using the output of one function as the input to another. Using this idea of composition of functions, we can sometimes find a function which will undo the action of another function, f . Such a function is called the inverse of f . For example, take a number x and multiply it by 6, giving 6 x . To get back to the original number x , you multiply 1 _ 6 by 6 x . This is an instance of undoing the action of a function. We next give the formal definition of the inverse of a function. Inverse of a Function Let f be a function. A function g is said to be the inverse function of f if the domain of g is equal to the range of f and, for every x in the domain of f and every y in the domain of g , g ( y ) = x if and only if f ( x ) = y The notation for the inverse function of f is f − 1 . Equivalently, f − 1 ( y ) = x if and only if f ( x ) = y The notation f − 1 does NOT mean 1 _ f . The idea of an inverse can be illustrated graphically as follows. Consider evaluating f − 1 (4) using the graph of a function f in Figure 1.

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  Precalculus

  Building Concepts and Connections 2E

  • Revathi Narasimhan(Author)
  • 2016(Publication Date)
  • XYZ Textbooks

    (Publisher)

  Consider the two functions f ( x ) = 2 x and g ( x ) = 2 x . a. Make a table of values of f ( x ) and g ( x ), with x ranging from − 1 to 4 in steps of 0.5. b. Find the interval(s) where 2 x < 2 x . c. Find the interval(s) where 2 x > 2 x . d. Using your table from part (a) as an aid, state what happens to the value of f ( x ) if x is increased by 1 unit. e. Using your table from part (a) as an aid, state what happens to the value of g ( x ) if x is increased by 1 unit. f. Using your answers from parts (c) and (d) as an aid, explain why the value of g ( x ) is increasing much faster than the value of f ( x ). 81. Explain why the function f ( x ) = 2 x has no vertical asymptotes. (Review Section 3.7.) 4 . 3 4.3 Logarithmic Functions 345 When you are given the output of an exponential function and asked to find the exponent, or the corresponding input, you are taking the inverse of the exponential function. This inverse function is called the logarithmic function . For instance, we can ask, “For what value of x is 2 x = 32?” The answer is x = 5. The exponent, 5, is called a logarithm , and the corresponding inverse functions is called the logarithmic function. Definition of Logarithm The following is the formal definition of a logarithm. Definition of Logarithm Let a > 0, a ≠ 1. If x > 0, then the logarithm of x with respect to base a is denoted by y = log a x and defined by y = log a x if and only if x = a y The number a is known as the base . Thus the functions f ( x ) = a x and g ( x ) = log a x are inverses of each other. That is, a log a x = x and log a a x = x This formal definition of a logarithm does not tell us how to calculate the value of log a x ; it simply gives a definition for such a number. Observations ■ The number denoted by log a x is defined to be the unique exponent y that satisfies the equation a y = x . ■ Substituting for y , the definition of the logarithm gives a y = a log a x = x Thus, a logarithm is an exponent .

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  Functions Modeling Change

  A Preparation for Calculus

  • Eric Connally, Deborah Hughes-Hallett, Andrew M. Gleason(Authors)
  • 2019(Publication Date)
  • Wiley

    (Publisher)

  For example, log(10 6 ) = 6 and 10 log 6 = 6. In general, For any  , log(10  ) =  and for  > 0, 10 log  = . Example 4 Evaluate without a calculator: (a) log ( 10 8.5 ) (b) 10 log 2.7 (c) 10 log(+3) Solution Using log(10  ) =  and 10 log  =  , we have: (a) log ( 10 8.5 ) = 8.5 (b) 10 log 2.7 = 2.7 (c) 10 log(+3) =  + 3 You can check the first two results on a calculator. Properties of Logarithms In Section 4.3, we solved the exponential equation 100 ⋅ 2  = 337,000,000 graphically. We now use logarithms and their properties to do so. These properties are justified on page 168. 166 Chapter 5 Logarithmic Functions Properties of the Common Logarithm • By definition,  = log  means 10  = . • In particular, log 1 = 0 and log 10 = 1. • The functions 10  and log  are inverses, so they “undo” each other: log(10  ) =  for all , 10 log  =  for  > 0. • For  and  both positive and any value of , log() = log  + log  log (   ) = log  − log  log(  ) =  ⋅ log . Example 5 Solve 100 ⋅ 2  = 337,000,000 for . Solution Dividing both sides of the equation by 100 gives 2  = 3,370,000. Taking logs of both sides gives log ( 2  ) = log(3,370,000). Since log(2  ) =  ⋅ log 2, we have  log 2 = log(3,370,000), so, solving for , we have  = log(3,370,000) log 2 = 21.684. In Example 2 on page 142, we estimated graphically that the fine faced by the city of Yonkers ex- ceeded the city’s annual budget between day 21 and day 22. The Natural Logarithm When  is used as the base for exponential functions, computations are easier with the use of another logarithm function, called log base . The log base  is used so frequently that it has its own notation: ln , read as the natural log of . We make the following definition: For  > 0, ln  is the power of  that gives  or, in symbols,  = ln  means   = , and  is called the natural logarithm of .

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