Technology & Engineering
Exponential and Logarithmic Functions
Exponential and logarithmic functions are mathematical tools commonly used in technology and engineering. Exponential functions represent rapid growth or decay, while logarithmic functions are used to solve for the exponent in an exponential equation. These functions are essential for modeling phenomena such as population growth, radioactive decay, and signal processing in technological and engineering applications.
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9 Key excerpts on "Exponential and Logarithmic Functions"
- Ron Larson(Author)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
Exponential and Logarithmic Functions 4.1 Exponential Functions 4.2 Natural Exponential Functions 4.3 Derivatives of Exponential Functions 4.4 Logarithmic Functions 4.5 Derivatives of Logarithmic Functions 4.6 Exponential Growth and Decay 4 Balance t A 2 4 6 8 10 12 4 P 3 P 2 P P Time (in years) Continuously Compounded Interest A = Pe rt (12, 4 P ) (0, P ) (6, 2 P ) Example 3 on page 297 shows how an exponential growth model can be used to find the annual interest rate of an account. Ammentorp | Dreamstime.com Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-300 252 Chapter 4 Exponential and Logarithmic Functions 4.1 Exponential Functions Use the properties of exponents to evaluate and simplify exponential expressions. Sketch the graphs of exponential functions. Exponential Functions You are already familiar with the behavior of algebraic functions such as f ( x ) = x 2 , g ( x ) = radical.alt2 x = x 1 H20862 2 , and h ( x ) = 1 x = x -1 each of which involves a variable raised to a constant power. By interchanging roles and raising a constant to a variable power, you obtain another important class of functions called exponential functions. Some examples are f ( x ) = 2 x , g ( x ) = parenleft.alt4 1 10 parenright.alt4 x = 1 10 x , and h ( x ) = 3 2 x = 9 x . In general, you can use any positive number a ≠ 1 as the base of an exponential function. Definition of Exponential Function If a > 0 and a ≠ 1, then the exponential function with base a is f ( x ) = a x . In the definition of an exponential function, the base a = 1 is excluded because it yields f ( x ) = 1 x = 1. This is a constant function, not an exponential function. When working with exponential functions, the properties of exponents, shown below, are useful. Properties of Exponents Let a and b be positive real numbers, and let x and y be real numbers. 1. a 0 = 1 2. a x a y = a x + y 3. a x a y = a x -y 4.
eBook - PDF- Ron Larson(Author)
- 2017(Publication Date)
- Cengage Learning EMEA(Publisher)
Graphical Reasoning Let f (x) = log a x and g(x) = a x , where a > 1. (a) Let a = 1.2 and use a graphing utility to graph the two functions in the same viewing window. What do you observe? Approximate any points of intersection of the two graphs. (b) Determine the value(s) of a for which the two graphs have one point of intersection. (c) Determine the value(s) of a for which the two graphs have two points of intersection. Copyright 2018 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. 5.5 Exponential and Logarithmic Models 396 Chapter 5 Exponential and Logarithmic Functions Recognize the five most common types of models involving Exponential and Logarithmic Functions. Use exponential growth and decay functions to model and solve real-life problems. Use Gaussian functions to model and solve real-life problems. Use logistic growth functions to model and solve real-life problems. Use logarithmic functions to model and solve real-life problems. Introduction The five most common types of mathematical models involving exponential functions and logarithmic functions are listed below. 1. Exponential growth model: y = ae bx , b > 0 2. Exponential decay model: y = ae -bx , b > 0 3. Gaussian model: y = ae -(x -b) 2 H20862c 4. Logistic growth model: y = a 1 + be -rx 5. Logarithmic models: y = a + b ln x, y = a + b log x The basic shapes of the graphs of these functions are shown below.
eBook - PDF- Cynthia Y. Young(Author)
- 2018(Publication Date)
- Wiley(Publisher)
The first step in this process consists of writing y 5 k x in an equivalent form using the natural logarithm. Use the properties of this section to write an equivalent form of the following implicitly defined functions. 103. y 5 2 x 104. y 5 4 x ⋅ 3 x11 3.4 Exponential and Logarithmic Equations 325 326 CHAPTER 3 Exponential and Logarithmic Functions SKILLS OBJECTIVES ■ ■ Use exponential growth functions to model populations, economic applications of assets, and initial spreading of diseases. ■ ■ Use exponential decay functions to model scenarios in medicine, economic depreciation of assets, archaeology, and forensic science. ■ ■ Represent distributions using a Gaussian (normal) model. ■ ■ Represent restricted growth with logistic models. ■ ■ Use logistic growth models to determine time to pay off debt. CONCEPTUAL OBJECTIVES ■ ■ Understand that exponential growth models assume “uninhibited” growth. ■ ■ Understand that the horizontal asymptote corresponds to a limiting value, often zero. ■ ■ Understand that the bell curve is used to represent standardized tests (IQ/SAT), as well as height/weight charts. ■ ■ Understand that the asymptote of a logistic model corresponds to the carrying capacity of the system. ■ ■ Understand why doubling your monthly payment will reduce the life of the loan by more than half. 3.5 EXPONENTIAL AND LOGARITHMIC MODELS NAME MODEL GRAPH APPLICATIONS Exponential growth ƒ1 t 2 5 ce kt k . 0 World populations, bacteria growth, appreciation, global spread of the HIV virus Exponential decay ƒ1 t 2 5 ce 2kt k .
eBook - PDF- R. Gustafson, Jeff Hughes(Authors)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
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. Chapter 5 Exponential and Logarithmic Functions 534 Getting Ready You should be able to complete these vocabulary and concept statements before you proceed to the practice exercises. Fill in the blanks. 1. The equation y 5 log b x is equivalent to . 2. The domain of a logarithmic function is the interval . 3. The of a logarithmic function is the interval s2`, `d. 4. b log b x 5 . 5. Because the exponential function is one-to-one, it has an function. 6. The inverse of an exponential function is called a function. 7. log b x is the to which b is raised to get x. 8. The y-axis is an of a graph of f sxd 5 log b x. 9. The graph of f sxd 5 log b x passes through the points and . 10. log 10 10 x 5 . 11. ln x means . 12. The domain of the function f sxd 5 ln x is the interval . 13. The range of the function f sxd 5 ln x is the interval . 14. The graph of f sxd 5 ln x has the as an asymptote 15. In the expression log x, the base is understood to be . 16. In the expression ln x, the base is understood to be . Practice Write each equation in logarithmic form. 17. 8 2 5 64 18. 10 3 5 1000 19. 4 22 5 1 16 20. 3 24 5 1 81 21. 1 1 2 2 25 5 32 22. 1 1 3 2 23 5 27 23. x y 5 z 24. m n 5 p Write each equation in exponential form. 25. log 3 81 5 4 26. log 7 7 5 1 27. log 1y2 1 8 5 3 28. log 1y5 1 5 0 29. log 4 1 64 5 23 30. log 6 1 36 5 22 31. log H9266 H9266 5 1 32. log 7 1 49 5 22 Find each value of x. 33. log 2 8 5 x 34. log 3 9 5 x 35. log 4 1 64 5 x 36. log 6 216 5 x 37. log 1y2 1 8 5 x 38. log 1y3 1 81 5 x 39. log 9 3 5 x 40. log 125 5 5 x 41. log 1y2 8 5 x 42. log 1y2 16 5 x 43. log 8 x 5 2 44. log 7 x 5 0 45. log 7 x 5 1 46. log 2 x 5 8 47.
eBook - PDF- 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.
No longer available |Learn more- James Stewart, Lothar Redlin, Saleem Watson(Authors)
- 2016(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. 374 CHAPTER 4 ■ Exponential and Logarithmic Functions EXAMPLE 4 ■ Comparing Different Rates of Population Growth In 2000 the population of the world was 6.1 billion, and the relative rate of growth was 1.4% per year. It is claimed that a rate of 1.0% per year would make a significant difference in the total population in just a few decades. Test this claim by estimating the population of the world in the year 2050 using a relative rate of growth of (a) 1.4% per year and (b) 1.0% per year. Graph the population functions for the next 100 years for the two relative growth rates in the same viewing rectangle. SOLUTION (a) By the exponential growth model we have n1 t 2 6.1e 0.014t where n1 t 2 is measured in billions and t is measured in years since 2000. Because the year 2050 is 50 years after 2000, we find n1 50 2 6.1e 0.014 1502 6.1e 0.7 < 12.3 The estimated population in the year 2050 is about 12.3 billion. (b) We use the function n1 t 2 6.1e 0.010t and find n1 50 2 6.1e 0.010 1502 6.1e 0.50 < 10.1 The estimated population in the year 2050 is about 10.1 billion. The graphs in Figure 4 show that a small change in the relative rate of growth will, over time, make a large difference in population size. Now Try Exercise 7 ■ EXAMPLE 5 ■ Expressing a Model in Terms of e A culture starts with 10,000 bacteria, and the number doubles every 40 minutes. (a) Find a function n1 t 2 n 0 2 t/a that models the number of bacteria after t hours. (b) Find a function n1 t 2 n 0 e rt that models the number of bacteria after t hours.
eBook - PDF- James Stewart, Lothar Redlin, Saleem Watson, , James Stewart, Lothar Redlin, Saleem Watson(Authors)
- 2015(Publication Date)
- Cengage Learning EMEA(Publisher)
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. 426 CHAPTER 4 ■ Exponential and Logarithmic Functions 89. Comparing Logarithms Which is larger, log 4 258 or log 5 620? 90. Inverse Function Find the inverse of the function f 1 x 2 2 3 x , and state its domain and range. 91. Compound Interest If $12,000 is invested at an interest rate of 10% per year, find the amount of the investment at the end of 3 years for each compounding method. (a) Semiannually (b) Monthly (c) Daily (d) Continuously 92. Compound Interest A sum of $5000 is invested at an inter-est rate of 8 1 2 % per year, compounded semiannually. (a) Find the amount of the investment after 1 1 2 years. (b) After what period of time will the investment amount to $7000? (c) If interest were compounded continously instead of semiannually, how long would it take for the amount to grow to $7000? 93. Compound Interest A money market account pays 5.2% annual interest, compounded daily. If $100,000 is invested in this account, how long will it take for the account to accumulate $10,000 in interest? 94. Compound Interest A retirement savings plan pays 4.5% interest, compounded continuously. How long will it take for an investment in this plan to double? 95–96 ■ APY Determine the annual percentage yield (APY) for the given nominal annual interest rate and compounding frequency. 95. 4.25%; daily 96. 3.2%; monthly 97. Cat Population The stray-cat population in a small town grows exponentially. In 1999 the town had 30 stray cats, and the relative growth rate was 15% per year. (a) Find a function that models the stray-cat population n 1 t 2 after t years.
eBook - PDF- Geoffrey Berresford, Andrew Rockett(Authors)
- 2015(Publication Date)
- Cengage Learning EMEA(Publisher)
Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 283 4.3 Differentiation of Logarithmic and Exponential Functions This function, multiplied by the constant 1 y 1 2 p , is the famous “bell-shaped curve” of statistics, used for predictions of many things from the IQs of newborn babies to stock prices. Solutions TO PRACTICE PROBLEMS 1. ƒ ( x ) d dx a ln x x b x 1 1 x 2 1 ln x x 2 1 ln x x 2 2. d dx ln ( x 3 5 x 1) 3 x 2 5 x 3 5 x 1 3. ƒ ( x ) e x xe x ƒ (1) e 1 1 e 1 2 e (which is approximately 2 2.718 5.436) 4. d dx e 1 x 3 y 3 d dx e 1 1 3 x 3 e 1 1 3 x 3 ( x 2 ) x 2 e 1 x 3 y 3 Derivative of ln x Derivative of x 4.3 Section Summary In this section we developed the formulas for differentiating logarithmic and exponential functions: d dx ln x 1 x d dx e x e x d dx ln f ( x ) f ( x ) f ( x ) d dx e f ( x ) e f ( x ) # f ( x ) The last of these has a useful special case when the exponent is of the form f ( x ) kx for a constant k : d dx e kx ke kx We used these formulas together with our other differentiation formulas (the Product, Quotient, and Chain Rules) to find rates of change and to optimize functions. Verification of the Differentiation Formulas Verification of the Power Rule for Arbitrary Powers On page 108 we proved the Power Rule d dx x n nx n 1 for positive integer expo-nents, and on page 226 we extended it to rational exponents. We can now show 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.
eBook - PDFApplied Calculus for the Managerial, Life, and Social Sciences
A Brief Approach
- Soo Tan(Author)
- 2014(Publication Date)
- Cengage Learning EMEA(Publisher)
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. 5.6 EXPONENTIAL FUNCTIONS AS MATHEMATICAL MODELS 399 5.6 Exercises 1. EXPONENTIAL GROWTH Given that a quantity Q 1 t 2 is described by the exponential growth function Q 1 t 2 5 300e 0.02t where t is measured in minutes, answer the following questions: a. What is the growth constant? b. What quantity is present initially? c. Complete the following table of values: t 0 10 20 100 1000 Q 2. EXPONENTIAL DECAY Given that a quantity Q 1 t 2 exhibiting exponential decay is described by the function Q 1 t 2 5 2000e 20.06t where t is measured in years, answer the following questions: a. What is the decay constant? b. What quantity is present initially? c. Complete the following table of values: t 0 5 10 20 100 Q 3. GROWTH OF BACTERIA The growth rate of the bacterium Escherichia coli, a common bacterium found in the human intestine, is proportional to its size. Under ideal laboratory conditions, when this bacterium is grown in a nutrient broth medium, the number of cells in a culture doubles approximately every 20 min. a. If the initial cell population is 100, determine the function Q 1 t 2 that expresses the exponential growth of the number of cells of this bacterium as a function of time t (in minutes). b. How long will it take for a colony of 100 cells to increase to a population of 1 million? c. If the initial cell population were 1000, how would this alter our model? 4. WORLD POPULATION The world population at the beginning of 1990 was 5.3 billion.
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