Exponential and Logarithmic Functions, Applications, and Models

Transcription

1 86 Eponential and Logarithmic Functions, Applications, and Models Eponential Functions In this section we introduce two new tpes of functions The first of these is the eponential function Eponential Function An eponential function with base b, where b and b, is a function of the form f() b, where is an real number

2 86 Eponential and Logarithmic Functions, Applications, and Models 5 Compare with the discussion in the tet Thus far, we have defined onl integer eponents In the definition of eponential function, we allow to take on an real number value B using methods not discussed in this book, epressions such as 9/7, 5, can be approimated A scientific or graphing calculator is capable of determining approimations for these numbers See the screen in the margin Notice that in the definition of eponential function, the base b is restricted to positive numbers, with b and 3 EXAMPLE The graphs of f, g, and h are shown in Figure 3(a), (b), and (c) In each case, a table of selected points is given The points are joined with a smooth curve, tpical of the graphs of eponential functions Notice that for each graph, the curve approaches but does not intersect the -ais For this reason, the -ais is called the horizontal asmptote of the graph f() _ f() = g() _ g() = ( _ ) h() h() = (a) (b) (c) g() = ( ) _ h() = Compare with Figure 3 f() = FIGURE 3 Eample illustrates the following facts about the graph of an eponential function Graph of f b The graph alwas will contain the point,, since b When b, the graph will rise from left to right (as in the eample for b and b ) When b, the graph will fall from left to right (as in the eample for b ) 3 The -ais is the horizontal asmptote The domain is, and the range is, Probabl the most important eponential function has the base e The number e is named after Leonhard Euler (77 783), and is approimatel It is an irrational number, and its value is approached b the epression n n as n takes on larger and larger values We write

3 6 CHAPTER 8 Graphs, Functions, and Sstems of Equations and Inequalities as n l, n n l e See the table that follows n Approimate Value of n n This table shows selected values for Y XX , 785,, 788 f() = e 3 e FIGURE 33 Powers of e can be approimated on a scientific or graphing calculator Some powers of e obtained on a calculator are e , e , e The graph of the function f e is shown in Figure 33 Applications of Eponential Functions A real-life application of eponential functions occurs in the computation of compound interest e e 3 f() = e 3 5 e The values for,, and are approimated in this split-screen graph of f e Compound Interest Formula Suppose that a principal of P dollars is invested at an annual interest rate r (in percent, epressed as a decimal), compounded n times per ear Then the amount A accumulated after t ears is given b the formula A P r nnt EXAMPLE Suppose that $ is invested at an annual rate of 8%, compounded quarterl (four times per ear) Find the total amount in the account after ten ears if no withdrawals are made Use the compound interest formula, with P, r 8, n, and t A P n r nt A 8 A

4 86 Eponential and Logarithmic Functions, Applications, and Models 7 Using a calculator with an eponential ke, 8 Multipl b to get A 8 There would be $8 in the account at the end of ten ears The compounding formula given earlier applies if the financial institution compounds interest for a finite number of compounding periods annuall Theoreticall, the number of compounding periods per ear can get larger and larger (quarterl, monthl, dail, etc), and if n is allowed to approach infinit, we sa that interest is compounded continuousl It can be shown that the formula for continuous compounding involves the number e Continuous Compounding Formula Suppose that a principal of P dollars is invested at an annual interest rate r (in percent, epressed as a decimal), compounded continuousl Then the amount A accumulated after t ears is given b the formula A Pe rt EXAMPLE 3 Suppose that $5 is invested at an annual rate of 65%, compounded continuousl Find the total amount in the account after four ears if no withdrawals are made The continuous compounding formula applies here, with P 5, r 65, and t A 5e6 Use the e ke on a calculator to find that e , so A There will be $6865 in the account after four ears The continuous compounding formula is an eample of an eponential growth function It can be shown that in situations involving growth or deca of a quantit, the amount or number present at time t can often be closel approimated b a function of the form A(t) A e kt, where A represents the amount or number present at time t, and k is a constant If k, there is eponential growth; if k, there is eponential deca Logarithmic Functions A Pe rt A 5e 65 Consider the eponential equation 3 8 Here we see that 3 is the eponent (or power) to which must be raised in order to obtain 8 The eponent 3 is called the logarithm to the base of 8, and this is written 3 log 8 In general, we have the following relationship For b, b, if b, then log b

5 8 CHAPTER 8 Graphs, Functions, and Sstems of Equations and Inequalities F() = log H() = log The following table illustrates the relationship between eponential equations and logarithmic equations Eponential Equation Logarithmic Equation 5 G() = log / Compare with Figure 3 on the net page Graphs of logarithmic functions with bases other than and e are accomplished with the use of the change-of-base rule from algebra: log a log ln log a ln a Alternativel, the can be drawn b using the capabilit that allows the user to obtain the graph of the inverse log / 6 log The concept of inverse functions (studied in more advanced algebra courses) leads us to the definition of the logarithmic function with base b Logarithmic Function log 3 8 A logarithmic function with base b, where b and b, is a function of the form g log b, where FOR FURTHER THOUGHT In the Februar 5, 989, issue of Parade, Carl Sagan related an oft-told legend involving eponential growth in the article The Secret of the Persian Chessboard Once upon a time, a Persian king wanted to please his eecutive officer, the Grand Vizier, with a gift of his choice The Grand Vizier eplained that he would like to be able to use his chessboard to accumulate wheat A single grain of wheat would be received for the first square on the board, two grains would be received for the second square, four grains for the third, and so on, doubling the number of grains for each of the 6 squares on the board This doubling procedure is an eample of eponential growth, and is defined b the eponential function f, where corresponds to the number of the chessboard square, and f represents the number of grains of wheat corresponding to that square How man grains of wheat would be accumulated? As unlikel as it ma seem, the number of grains would total 85 quintillion! Even with toda s methods of production, this amount would take 5 ears to produce The Grand Vizier evidentl knew his mathematics For Group Discussion If a lil pad doubles in size each da, and it covers a pond after eight das of growth, when is the pond half covered? One-fourth covered? Have each member of the class estimate the answer to this problem If ou earn on Januar, on Januar, on Januar 3, 8 on Januar, and so on, doubling our salar each da, how much will ou earn during the month? As a calculator eercise, determine the answer and see who was able to estimate the answer most accuratel

6 86 Eponential and Logarithmic Functions, Applications, and Models 9 The graph of the function g log b can be found b interchanging the roles of and in the function f b Geometricall, this is accomplished b reflecting the graph of f b about the line EXAMPLE The graphs of F log, G log /, and H log are shown in Figure 3(a), (b), and (c) In each case, a table of selected points is given These points were obtained b interchanging the roles of and in the tables of points given in Figure 3 The points are joined with a smooth curve, tpical of the graphs of logarithmic functions Notice that for each graph, the curve approaches but does not intersect the -ais For this reason, the -ais is called the vertical asmptote of the graph _ F() F() = log 3 _ G() G() = log / 3 H() H() = log (a) (b) (c) FIGURE 3 g() = ln Eample illustrates the following facts about the graph of a logarithmic function 5 Notice that when e 7888, ln e Graph of g log b The graph will alwas contain the point,, since log b When b, the graph will rise from left to right, from the fourth quadrant to the first (as in the eample for b and b ) When b, the graph will fall from left to right, from the first quadrant to the fourth (as in the eample for b ) 3 The -ais is the vertical asmptote The domain is, and the range is, g() = ln 6 8 FIGURE 35 One of the most important logarithmic functions is the function with base e If we interchange the roles of and in the graph of f e (Figure 33), we obtain the graph of g log e There is a special smbol for log e : it is ln That is, Figure 35 shows the graph of ln log e g ln, called the natural logarithmic function

7 5 CHAPTER 8 Graphs, Functions, and Sstems of Equations and Inequalities The number e is named in honor of Leonhard Euler (77 783), the prolific Swiss mathematician The value of e can be epressed as an infinite series e 3 3 It can also be epressed in two was using continued fractions e e 3 3 The epression ln e k is the eponent to which the base e must be raised in order to obtain e k There is onl one such number that will do this, and it is k itself Therefore, we make the following important statement For all real numbers k, ln e k k This fact is used in the eamples that follow EXAMPLE 5 Suppose that a certain amount P is invested at an annual rate of 65%, compounded continuousl How long will it take for the amount to triple? We wish to find the value of t in the continuous compounding formula that will make the amount A equal to 3P (since we want the initial investment, P, to triple) We use the formula A Pe rt, with 3P substituted for A and 65 substituted for r A Pe rt 3P Pe 65t 3 e 65t Now take the natural logarithm of both sides ln 3 ln e 65t ln 3 65t Solve for t b dividing both sides b 65 Divide both sides b P Use the fact that ln e k k t ln 3 65 A calculator shows that ln Dividing this b 65 gives t 69 to the nearest tenth Therefore, it would take about 69 ears for an initial investment P to triple under the given conditions Eponential Models EXAMPLE 6 The greenhouse effect refers to the phenomenon whereb emissions of gases such as carbon dioide, methane, and chlorofluorocarbons (CFCs) have the potential to alter the climate of the earth and destro the ozone laer Concentrations of CFC-, used in refrigeration technolog, in parts per billion (ppb) can be modeled b the eponential function defined b f 8e, where represents 99, represents 99, and so on Use this function to approimate the concentration in 998

8 86 Eponential and Logarithmic Functions, Applications, and Models 5 Since represents 99, 8 represents 998 Evaluate f8 using a calculator f8 8e (8 8e 3 66 In 998, the concentration of CFC- was about 66 ppb Radioactive materials disintegrate according to eponential deca functions The half-life of a quantit that decas eponentiall is the amount of time that it takes for an initial amount to deca to half its initial value EXAMPLE 7 Carbon is a radioactive form of carbon that is found in all living plants and animals After a plant or animal dies, the radiocarbon disintegrates Scientists determine the age of the remains b comparing the amount of carbon present with the amount found in living plants and animals The amount of carbon present after t ears is modeled b the eponential equation e6t, where represents the initial amount What is the half-life of carbon? To find the half-life, let in the equation e 6t e6t ln 6t t 57 The half-life of carbon is about 57 ears Use a calculator FOR FURTHER THOUGHT So what does that log button on our scientific calculator mean? We will answer that question shortl But let s back up to 6 when the British mathematician Henr Briggs published what was, at that time, the greatest breakthrough ever for aiding computation: tables of common logarithms A common, or base ten logarithm, of a positive number is the eponent to which must be raised in order to obtain Smbolicall, we simpl write log to denote log Since logarithms are eponents, their properties are the same as those of eponents These properties allowed users of tables of common logarithms to multipl b adding, divide b subtracting, raise to powers b multipling, and take roots b dividing For eample, to multipl 53 b 967, the user would find log 53 in the table, then find log 967 in the table, and add these values Then, b reading the table in reverse, the user would find the antilog of the sum in order to find the product of the original two numbers More complicated operations were possible as well (continued)

9 5 CHAPTER 8 Graphs, Functions, and Sstems of Equations and Inequalities The photo here shows a homework problem that someone was asked to do on Ma 5, 93 It was found on a well-preserved sheet of paper stuck inside an old mathematics tet purchased in a used-book store in New Orleans Aren t ou glad that we now use calculators? So wh include a base ten logarithm ke on a scientific calculator? Common logarithms still are used in certain applications in science For eample, the ph of a substance is the negative of the common logarithm of the hdronium ion concentration in moles per liter The ph value is a measure of the acidit or alkalinit of a solution If ph 7, the solution is alkaline; if ph 7, it is acidic The Richter scale, used to measure the intensit of earthquakes, is based on logarithms, and sound intensities, measured in decibels, also have a logarithmic basis For Group Discussion Divide the class into small groups, and use the log ke of a scientific calculator to multipl 583 b 96 Remember that once the logarithms have been added, it is necessar then to raise to that power to obtain the product The ke will be required for this Ask someone who studied computation with logarithms in an algebra or trigonometr course before the advent of scientific calculators what it was like Then report their recollections to the class 3 What is a slide rule? Have someone in the class find one in an attic or closet, and bring it to class The slide rule was a popular tool for calculation through the 96s