Exponential and Logarithmic Functions

Eponential and Logarithmic Functions 6 Figure Electron micrograph of E. Coli bacteria (credit: Mattosaurus, Wikimedia Commons) CHAPTER OUTLINE 6. Eponential Functions 6. Logarithmic Properties 6. Graphs of Eponential Functions 6.6 Eponential and Logarithmic Equations 6. Logarithmic Functions 6.7 Eponential and Logarithmic Models 6. Graphs of Logarithmic Functions 6.8 Fitting Eponential Models to Data Introduction Focus in on a square centimeter of our skin. Look closer. Closer still. If ou could look closel enough, ou would see hundreds of thousands of microscopic organisms. The are bacteria, and the are not onl on our skin, but in our mouth, nose, and even our intestines. In fact, the bacterial cells in our bod at an given moment outnumber our own cells. But that is no reason to feel bad about ourself. While some bacteria can cause illness, man are health and even essential to the bod. Bacteria commonl reproduce through a process called binar fission, during which one bacterial cell splits into two. When conditions are right, bacteria can reproduce ver quickl. Unlike humans and other comple organisms, the time required to form a new generation of bacteria is often a matter of minutes or hours, as opposed to das or ears. [6] For simplicit s sake, suppose we begin with a culture of one bacterial cell that can divide ever hour. Table shows the number of bacterial cells at the end of each subsequent hour. We see that the single bacterial cell leads to over one thousand bacterial cells in just ten hours! And if we were to etrapolate the table to twent-four hours, we would have over 6 million! Hour 0 6 7 8 9 0 Bacteria 8 6 6 8 6 0 Table In this chapter, we will eplore eponential functions, which can be used for, among other things, modeling growth patterns such as those found in bacteria. We will also investigate logarithmic functions, which are closel related to eponential functions. Both tpes of functions have numerous real-world applications when it comes to modeling and interpreting data. 6 Todar, PhD, Kenneth. Todar s Online Te tbook of Bacteriolog. http://te tbookofbacteriolog.net/growth_.html. Download for free at http://cn.org/contents/9b08c9-07f-0-9f8-d6ad9970d@.9 6

SECTION 6. GRAPHS OF LOGARITHMIC FUNCTIONS 99 LEARNING OBJECTIVES In this section, ou will: Identif the domain of a logarithmic function. Graph logarithmic functions. 6. GRAPHS OF LOGARITHMIC FUNCTIONS In Graphs of Eponential Functions, we saw how creating a graphical representation of an eponential model gives us another laer of insight for predicting future events. How do logarithmic graphs give us insight into situations? Because ever logarithmic function is the inverse function of an eponential function, we can think of ever output on a logarithmic graph as the input for the corresponding inverse eponential equation. In other words, logarithms give the cause for an effect. To illustrate, suppose we invest $,00 in an account that offers an annual interest rate of %, compounded continuousl. We alread know that the balance in our account for an ear t can be found with the equation A = 00e 0.0t. But what if we wanted to know the ear for an balance? We would need to create a corresponding new function b interchanging the input and the output; thus we would need to create a logarithmic model for this situation. B graphing the model, we can see the output (ear) for an input (account balance). For instance, what if we wanted to know how man ears it would take for our initial investment to double? Figure shows this point on the logarithmic graph. Years 0 8 6 0 8 6 0 Logarithmic Model Showing Years as a Function of the Balance in the Account 00,000 The balance reaches $,000 near ear,00,000,00,000,00,000,00,000,00 6,000 Account balance Figure In this section we will discuss the values for which a logarithmic function is defined, and then turn our attention to graphing the famil of logarithmic functions. Finding the Domain of a Logarithmic Function Before working with graphs, we will take a look at the domain (the set of input values) for which the logarithmic function is defined. Recall that the eponential function is defined as = b for an real number and constant b > 0, b, where The domain of is (, ). The range of is (0, ). In the last section we learned that the logarithmic function = log b () is the inverse of the eponential function = b. So, as inverse functions: The domain of = log b () is the range of = b : (0, ). The range of = log b () is the domain of = b : (, ). Download for free at http://cn.org/contents/9b08c9-07f-0-9f8-d6ad9970d@.9

00 CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS Transformations of the parent function = log b () behave similarl to those of other functions. Just as with other parent functions, we can appl the four tpes of transformations shifts, stretches, compressions, and reflections to the parent function without loss of shape. In Graphs of Eponential Functions we saw that certain transformations can change the range of = b. Similarl, appling transformations to the parent function = log b () can change the domain. When finding the domain of a logarithmic function, therefore, it is important to remember that the domain consists onl of positive real numbers. That is, the argument of the logarithmic function must be greater than zero. For eample, consider f () = log ( ). This function is defined for an values of such that the argument, in this case, is greater than zero. To find the domain, we set up an inequalit and solve for : > 0 > Add. Show the argument greater than zero. >. Divide b. In interval notation, the domain of f () = log ( ) is (., ). How To Given a logarithmic function, identif the domain.. Set up an inequalit showing the argument greater than zero.. Solve for.. Write the domain in interval notation. Eample Identifing the Domain of a Logarithmic Shift What is the domain of f () = log ( + )? Solution The logarithmic function is defined onl when the input is positive, so this function is defined when + > 0. Solving this inequalit, + > 0 The input must be positive. > Subtract. The domain of f () = log ( + ) is (, ). Tr It # What is the domain of f () = log ( ) +? Eample Identifing the Domain of a Logarithmic Shift and Reflection What is the domain of f () = log( )? Solution The logarithmic function is defined onl when the input is positive, so this function is defined when > 0. Solving this inequalit, > 0 The input must be positive. > Subtract. < Divide b and switch the inequalit. The domain of f () = log( ) is (, _ ). Download for free at http://cn.org/contents/9b08c9-07f-0-9f8-d6ad9970d@.9

SECTION 6. GRAPHS OF LOGARITHMIC FUNCTIONS 0 Tr It # What is the domain of f () = log( ) +? Graphing Logarithmic Functions Now that we have a feel for the set of values for which a logarithmic function is defined, we move on to graphing logarithmic functions. The famil of logarithmic functions includes the parent function = log b () along with all its transformations: shifts, stretches, compressions, and reflections. We begin with the parent function = log b (). Because ever logarithmic function of this form is the inverse of an eponential function with the form = b, their graphs will be reflections of each other across the line =. To illustrate this, we can observe the relationship between the input and output values of = and its equivalent = log () in Table. 0 = _ 8 8 log () = 0 Table Using the inputs and outputs from Table, we can build another table to observe the relationship between points on the graphs of the inverse functions f () = and g() = log (). See Table. f () = (, _ 8) (, _ ) (, _ ) g() = log () ( _ 8, ) ( _, ) ( _, ) (0, ) (, ) (, ) (, 8) (, 0) (, ) (, ) (8, ) Table As we d epect, the - and -coordinates are reversed for the inverse functions. Figure shows the graph of f and g. f () = = g() = log () Figure Notice that the graphs of f () = and g() = log () are reflections about the line =. Observe the following from the graph: f () = has a -intercept at (0, ) and g() = log () has an -intercept at (, 0). The domain of f () =, (, ), is the same as the range of g() = log (). The range of f () =, (0, ), is the same as the domain of g() = log (). Download for free at http://cn.org/contents/9b08c9-07f-0-9f8-d6ad9970d@.9

0 CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS characteristics of the graph of the parent function, f () = log b () For an real number and constant b > 0, b, we can see the following characteristics in the graph of f () = log b (): one-to-one function vertical asmptote: = 0 domain: (0, ) range: (, ) -intercept: (, 0) and ke point (b, ) -intercept: none increasing if b > decreasing if 0 < b < See Figure. = 0 Figure shows how changing the base b in f () = log b () can affect the graphs. Observe that the graphs compress verticall as the value of the base increases. (Note: recall that the function ln() has base e.78.) f() (, 0) f() = log b () b > (b, ) Figure = 0 f() 0 8 6 (b, ) (, 0) = 0 f () = log b () 0 < b < log () ln() log() 6 8 0 Figure The graphs of three logarithmic functions with different bases, all greater than. How To Given a logarithmic function with the form f () = log b (), graph the function.. Draw and label the vertical asmptote, = 0.. Plot the -intercept, (, 0).. Plot the ke point (b, ).. Draw a smooth curve through the points.. State the domain, (0, ), the range, (, ), and the vertical asmptote, = 0. Eample Graphing a Logarithmic Function with the Form f ( ) = log b ( ). Graph f () = log (). State the domain, range, and asmptote. Solution Before graphing, identif the behavior and ke points for the graph. Since b = is greater than one, we know the function is increasing. The left tail of the graph will approach the vertical asmptote = 0, and the right tail will increase slowl without bound. The -intercept is (, 0). The ke point (, ) is on the graph. We draw and label the asmptote, plot and label the points, and draw a smooth curve through the points (see Figure ). Download for free at http://cn.org/contents/9b08c9-07f-0-9f8-d6ad9970d@.9

SECTION 6. GRAPHS OF LOGARITHMIC FUNCTIONS 0 f () 0 8 6 = 0 (, ) f () = log () 6 8 0 (, 0) Figure The domain is (0, ), the range is (, ), and the vertical asmptote is = 0. Tr It # Graph f () = log _ (). State the domain, range, and asmptote. Graphing Transformations of Logarithmic Functions As we mentioned in the beginning of the section, transformations of logarithmic graphs behave similarl to those of other parent functions. We can shift, stretch, compress, and reflect the parent function = log b () without loss of shape. Graphing a Horizontal Shift of f ( ) = log b ( ) When a constant c is added to the input of the parent function f () = log b (), the result is a horizontal shift c units in the opposite direction of the sign on c. To visualize horizontal shifts, we can observe the general graph of the parent function f () = log b () and for c > 0 alongside the shift left, g( ) = log b ( + c), and the shift right, h() = log b ( c). See Figure 6. Shift left g () = log b ( + c) = c = 0 g() = log b ( + c) = 0 Shift right h() = log b ( c) = c f () = log b () (b c, ) f () = log b () (b, ) (, 0) ( c, 0) (b, ) (b + c, ) (, 0) ( + c, 0) h() = log b ( c) The asmptote changes to = c. The domain changes to ( c, ). The range remains (, ). Figure 6 The asmptote changes to = c. The domain changes to (c, ). The range remains (, ). horizontal shifts of the parent function = log b () For an constant c, the function f () = log b ( + c) shifts the parent function = log b () left c units if c > 0. shifts the parent function = log b () right c units if c < 0. has the vertical asmptote = c. has domain ( c, ). has range (, ). Download for free at http://cn.org/contents/9b08c9-07f-0-9f8-d6ad9970d@.9

0 CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS How To Given a logarithmic function with the form f () = log b ( + c), graph the translation.. Identif the horizontal shift: a. If c > 0, shift the graph of f () = log b () left c units. b. If c < 0, shift the graph of f () = log b () right c units.. Draw the vertical asmptote = c.. Identif three ke points from the parent function. Find new coordinates for the shifted functions b subtracting c from the coordinate.. Label the three points.. The domain is ( c, ), the range is (, ), and the vertical asmptote is = c. Eample Graphing a Horizontal Shift of the Parent Function = log b ( ) Sketch the horizontal shift f () = log ( ) alongside its parent function. Include the ke points and asmptotes on the graph. State the domain, range, and asmptote. Solution Since the function is f () = log ( ), we notice + ( ) =. Thus c =, so c < 0. This means we will shift the function f () = log () right units. The vertical asmptote is = ( ) or =. Consider the three ke points from the parent function, _ (, ), (, 0), and (, ). The new coordinates are found b adding to the coordinates. Label the points _ ( 7, ), (, 0), and (, ). The domain is (, ), the range is (, ), and the vertical asmptote is =. (, 0) 6 7 8 9 = 0 = (, ) (, 0) (, ) Figure 7 = log () f () = log ( ) Tr It # Sketch a graph of f () = log ( + ) alongside its parent function. Include the ke points and asmptotes on the graph. State the domain, range, and asmptote. Graphing a Vertical Shift of = log b ( ) When a constant d is added to the parent function f () = log b (), the result is a vertical shift d units in the direction of the sign on d. To visualize vertical shifts, we can observe the general graph of the parent function f () = log b () alongside the shift up, g () = log b () + d and the shift down, h() = log b () d. See Figure 8. Download for free at http://cn.org/contents/9b08c9-07f-0-9f8-d6ad9970d@.9

SECTION 6. GRAPHS OF LOGARITHMIC FUNCTIONS 0 Shift up g () = log b () + d Shift down h() = log b () d = 0 g() = log b () + d = 0 f () = log b () (b d, ) (b d, 0) (b, ) (, 0) f () = log b () (b, ) (, 0) (b +d, ) (b d, 0) h() = log b () d The asmptote remains = 0. The domain remains to (0, ). The range remains (, ). The asmptote remains = 0. The domain remains to (0, ). The range remains (, ). Figure 8 vertical shifts of the parent function = log b () For an constant d, the function f () = log b () + d shifts the parent function = log b () up d units if d > 0. shifts the parent function = log b () down d units if d < 0. has the vertical asmptote = 0. has domain (0, ). has range (, ). How To Given a logarithmic function with the form f () = log b () + d, graph the translation.. Identif the vertical shift: a. If d > 0, shift the graph of f () = log b () up d units. b. If d < 0, shift the graph of f () = log b () down d units.. Draw the vertical asmptote = 0.. Identif three ke points from the parent function. Find new coordinates for the shifted functions b adding d to the coordinate.. Label the three points.. The domain is (0, ), the range is (, ), and the vertical asmptote is = 0. Eample Graphing a Vertical Shift of the Parent Function = log b () Sketch a graph of f () = log () alongside its parent function. Include the ke points and asmptote on the graph. State the domain, range, and asmptote. Solution Since the function is f () = log (), we will notice d =. Thus d < 0. This means we will shift the function f () = log () down units. The vertical asmptote is = 0. Download for free at http://cn.org/contents/9b08c9-07f-0-9f8-d6ad9970d@.9

06 CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS Consider the three ke points from the parent function, _ (, ), (, 0), and (, ). The new coordinates are found b subtracting from the coordinates. Label the points _ (, ), (, ), and (, ). The domain is (0, ), the range is (, ), and the vertical asmptote is = 0. (, 0) (, ) = log () 6 7 8 9 (, ) f () = log ( ) (, ) = 0 Figure 9 The domain is (0, ), the range is (, ), and the vertical asmptote is = 0. Tr It # Sketch a graph of f () = log () + alongside its parent function. Include the ke points and asmptote on the graph. State the domain, range, and asmptote. Graphing Stretches and Compressions of = log b ( ) When the parent function f () = log b () is multiplied b a constant a > 0, the result is a vertical stretch or compression of the original graph. To visualize stretches and compressions, we set a > and observe the general graph of the parent function f () = log b () alongside the vertical stretch, g () = alog b () and the vertical compression, h() = _ a log (). b See Figure 0. Vertical Stretch g () = alog b (), a > g() = alog b () = 0 Vertical Compression h() = _ a log b (), a > = 0 /a (b, ) (b, ) (, 0) f() = log b () (b, ) (, 0) f() = log b () h() = a log b () (b a, ) The asmptote remains = 0. The -intercept remains (, 0). The domain remains (0, ). The range remains (, ). The asmptote remains = 0. The -intercept remains (, 0). The domain remains (0, ). The range remains (, ). Figure 0 Download for free at http://cn.org/contents/9b08c9-07f-0-9f8-d6ad9970d@.9

SECTION 6. GRAPHS OF LOGARITHMIC FUNCTIONS 07 vertical stretches and compressions of the parent function = log b () For an constant a >, the function f () = alog b () stretches the parent function = log b () verticall b a factor of a if a >. compresses the parent function = log b () verticall b a factor of a if 0 < a <. has the vertical asmptote = 0. has the -intercept (, 0). has domain (0, ). has range (, ). How To Given a logarithmic function with the form f () = alog b (), a > 0, graph the translation.. Identif the vertical stretch or compressions: a. If a >, the graph of f () = log b () is stretched b a factor of a units. b. If a <, the graph of f () = log b () is compressed b a factor of a units.. Draw the vertical asmptote = 0.. Identif three ke points from the parent function. Find new coordinates for the shifted functions b multipling the coordinates b a.. Label the three points.. The domain is (0, ), the range is (, ), and the vertical asmptote is = 0. Eample 6 Graphing a Stretch or Compression of the Parent Function = log b ( ) Sketch a graph of f () = log () alongside its parent function. Include the ke points and asmptote on the graph. State the domain, range, and asmptote. Solution Since the function is f () = log (), we will notice a =. This means we will stretch the function f () = log () b a factor of. The vertical asmptote is = 0. Consider the three ke points from the parent function, _ (, ), (, 0), and (, ). The new coordinates are found b multipling the coordinates b. Label the points _ (, ), (, 0), and (, ). The domain is (0, ), the range is (, ), and the vertical asmptote is = 0. See Figure. = 0 (, ) f () = log () (, ) = log () (, ) 6 7 8 9 (, 0) Figure Download for free at http://cn.org/contents/9b08c9-07f-0-9f8-d6ad9970d@.9

08 CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS Tr It #6 Sketch a graph of f () = _ log () alongside its parent function. Include the ke points and asmptote on the graph. State the domain, range, and asmptote. Eample 7 Combining a Shift and a Stretch Sketch a graph of f () = log( + ). State the domain, range, and asmptote. Solution Remember: what happens inside parentheses happens first. First, we move the graph left units, then stretch the function verticall b a factor of, as in Figure. The vertical asmptote will be shifted to =. The -intercept will be (, 0). The domain will be (, ). Two points will help give the shape of the graph: (, 0) and (8, ). We chose = 8 as the -coordinate of one point to graph because when = 8, + = 0, the base of the common logarithm. = Figure = log( + ) = log( + ) = log() The domain is (, ), the range is (, ), and the vertical asmptote is =. Tr It #7 Sketch a graph of the function f () = log( ) +. State the domain, range, and asmptote. Graphing Reflections of f ( ) = logb( ) When the parent function f () = log b () is multiplied b, the result is a reflection about the -ais. When the input is multiplied b, the result is a reflection about the -ais. To visualize reflections, we restrict b >, and observe the general graph of the parent function f () = log b () alongside the reflection about the -ais, g() = log b () and the reflection about the -ais, h() = log b ( ). Reflection about the -ais g () = log b (), b > Reflection about the -ais h() = log b ( ), b > = 0 = 0 (b, ) (, 0) (b, ) f () = log b () g() = log b () h() = log b ( ) ( b, ) (, 0) f () = log b () (, 0) (b, ) The reflected function is decreasing as moves from zero to infinit. The asmptote remains = 0. The -intercept remains (, 0). The ke point changes to (b, ). The domain remains (0, ). The range remains (, ). Download for free at http://cn.org/contents/9b08c9-07f-0-9f8-d6ad9970d@.9 Figure The reflected function is decreasing as moves from infinit to zero. The asmptote remains = 0. The -intercept remains (, 0). The ke point changes to ( b, ). The domain changes to (, 0). The range remains (, ).

SECTION 6. GRAPHS OF LOGARITHMIC FUNCTIONS 09 reflections of the parent function = log b () The function f () = log b () reflects the parent function = log b () about the -ais. has domain, (0, ), range, (, ), and vertical asmptote, = 0, which are unchanged from the parent function. The function f () = log b ( ) reflects the parent function = log b () about the -ais. has domain (, 0). has range, (, ), and vertical asmptote, = 0, which are unchanged from the parent function. How To Given a logarithmic function with the parent function f () = log b (), graph a translation. If f () = log b () If f () = log b ( ). Draw the vertical asmptote, = 0.. Draw the vertical asmptote, = 0.. Plot the -intercept, (, 0).. Plot the -intercept, (, 0).. Reflect the graph of the parent function f () = log b () about the -ais.. Reflect the graph of the parent function f () = log b () about the -ais.. Draw a smooth curve through the points.. Draw a smooth curve through the points.. State the domain, (0, ), the range, (, ), and the vertical asmptote = 0. Table. State the domain, (, 0), the range, (, ), and the vertical asmptote = 0. Eample 8 Graphing a Reflection of a Logarithmic Function Sketch a graph of f () = log( ) alongside its parent function. Include the ke points and asmptote on the graph. State the domain, range, and asmptote. Solution Before graphing f () = log( ), identif the behavior and ke points for the graph. Since b = 0 is greater than one, we know that the parent function is increasing. Since the input value is multiplied b, f is a reflection of the parent graph about the -ais. Thus, f () = log( ) will be decreasing as moves from negative infinit to zero, and the right tail of the graph will approach the vertical asmptote = 0. The -intercept is (, 0). We draw and label the asmptote, plot and label the points, and draw a smooth curve through the points. = 0 (0, 0) f () = log( ) = log() (0, 0) (, 0) (, 0) Figure The domain is (, 0), the range is (, ), and the vertical asmptote is = 0. Tr It #8 Graph f () = log( ). State the domain, range, and asmptote. Download for free at http://cn.org/contents/9b08c9-07f-0-9f8-d6ad9970d@.9

0 CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS How To Given a logarithmic equation, use a graphing calculator to approimate solutions.. Press [Y=]. Enter the given logarithm equation or equations as Y = and, if needed, Y =.. Press [GRAPH] to observe the graphs of the curves and use [WINDOW] to find an appropriate view of the graphs, including their point(s) of intersection.. To find the value of, we compute the point of intersection. Press [ND] then [CALC]. Select intersect and press [ENTER] three times. The point of intersection gives the value of, for the point(s) of intersection. Eample 9 Approimating the Solution of a Logarithmic Equation Solve ln() + = ln( ) graphicall. Round to the nearest thousandth. Solution Press [Y=] and enter ln() + net to Y =. Then enter ln( ) net to Y =. For a window, use the values 0 to for and 0 to 0 for. Press [GRAPH]. The graphs should intersect somewhere a little to right of =. For a better approimation, press [ND] then [CALC]. Select [: intersect] and press [ENTER] three times. The -coordinate of the point of intersection is displaed as.897. (Your answer ma be different if ou use a different window or use a different value for Guess?) So, to the nearest thousandth,.9. Tr It #9 Solve log( + ) = log() graphicall. Round to the nearest thousandth. Summarizing Translations of the Logarithmic Function Now that we have worked with each tpe of translation for the logarithmic function, we can summarize each in Table to arrive at the general equation for translating eponential functions. Translations of the Parent Function = log b () Translation Shift Horizontall c units to the left Verticall d units up Stretch and Compress Stretch if a > Compression if a < Reflect about the -ais Reflect about the -ais General equation for all translations translations of logarithmic functions Table Form = log b ( + c) + d = alog b () = log b () = log b ( ) = alog b ( + c) + d All translations of the parent logarithmic function, = log b (), have the form f () = alog b ( + c) + d where the parent function, = log b (), b >, is shifted verticall up d units. shifted horizontall to the left c units. stretched verticall b a factor of a if a > 0. compressed verticall b a factor of a if 0 < a <. reflected about the -ais when a < 0. For f () = log( ), the graph of the parent function is reflected about the -ais. Download for free at http://cn.org/contents/9b08c9-07f-0-9f8-d6ad9970d@.9

SECTION 6. GRAPHS OF LOGARITHMIC FUNCTIONS Eample 0 Finding the Vertical Asmptote of a Logarithm Graph What is the vertical asmptote of f () = log ( + ) +? Solution The vertical asmptote is at =. Analsis The coefficient, the base, and the upward translation do not affect the asmptote. The shift of the curve units to the left shifts the vertical asmptote to =. Tr It #0 What is the vertical asmptote of f () = + ln( )? Eample Finding the Equation from a Graph Find a possible equation for the common logarithmic function graphed in Figure. f() 6 7 Figure Solution This graph has a vertical asmptote at = and has been verticall reflected. We do not know et the vertical shift or the vertical stretch. We know so far that the equation will have form: f () = alog( + ) + k It appears the graph passes through the points (, ) and (, ). Substituting (, ), Ne t, substituting in (, ), = alog( + ) + k Substitute (, ). = alog() + k Arithmetic. = k log() = 0. = alog( + ) + = alog() a = log() This gives us the equation f () = log( + ) +. log() Analsis Figure. _ Plug in (, ). Arithmetic. Solve for a. You can verif this answer b comparing the function values in Table with the points on the graph in 0 f () 0 0.896.9 6 7 8 f ().80.807.699.9 Table Download for free at http://cn.org/contents/9b08c9-07f-0-9f8-d6ad9970d@.9

CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS Tr It # Give the equation of the natural logarithm graphed in Figure 6. f() Figure 6 Q & A Is it possible to tell the domain and range and describe the end behavior of a function just b looking at the graph? Yes, if we know the function is a general logarithmic function. For eample, look at the graph in Figure 6. The graph approaches = (or thereabouts) more and more closel, so = is, or is ver close to, the vertical asmptote. It approaches from the right, so the domain is all points to the right, { > }. The range, as with all general logarithmic functions, is all real numbers. And we can see the end behavior because the graph goes down as it goes left and up as it goes right. The end behavior is that as +, f () and as, f (). Access these online resources for additional instruction and practice with graphing logarithms. Graph an Eponential Function and Logarithmic Function (http://openstacollege.org/l/grapheplog) Match Graphs with Eponential and Logarithmic Functions (http://openstacollege.org/l/matcheplog) Find the Domain of Logarithmic Functions (http://openstacollege.org/l/domainlog) Download for free at http://cn.org/contents/9b08c9-07f-0-9f8-d6ad9970d@.9

SECTION 6. SECTION EXERCISES 6. SECTION EXERCISES VERBAL. The inverse of ever logarithmic function is an eponential function and vice-versa. What does this tell us about the relationship between the coordinates of the points on the graphs of each?. What tpe(s) of translation(s), if an, affect the domain of a logarithmic function?. What tpe(s) of translation(s), if an, affect the range of a logarithmic function?. Consider the general logarithmic function f () = log b (). Wh can t be zero?. Does the graph of a general logarithmic function have a horizontal asmptote? Eplain. ALGEBRAIC For the following eercises, state the domain and range of the function. 6. f () = log ( + ) 7. h() = ln ( _ ) 8. g() = log ( + 9) 9. h() = ln( + 7) 0. f () = log ( ) For the following eercises, state the domain and the vertical asmptote of the function.. f () = log b ( ). g() = ln( ). f () = log( + ). f () = log( ) +. g() = ln( + 9) 7 For the following eercises, state the domain, vertical asmptote, and end behavior of the function. 6. f () = ln( ) 7. f () = log ( _ 9. g() = ln( + 6) 0. f () = log ( ) + 6 7) 8. h() = log( ) + For the following eercises, state the domain, range, and - and -intercepts, if the eist. If the do not eist, write DNE.. h() = log ( ) +. f () = log( + 0) +. g() = ln( ). f () = log ( + ). h() = ln() 9 GRAPHICAL For the following eercises, match each function in Figure 7 with the letter corresponding to its graph. A 6. 7. d() = log() 0 B C D E 8. f () = ln() 9. g() = log () 0. h() = log (). j() = log () Figure 7 Download for free at http://cn.org/contents/9b08c9-07f-0-9f8-d6ad9970d@.9

CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS For the following eercises, match each function in Figure 8 with the letter corresponding to its graph. 6 6 Figure 8 A B C.. f () = log _ (). g() = log (). h() = log _ () For the following eercises, sketch the graphs of each pair of functions on the same ais. 6. f () = log() and g() = 0 7. f () = log() and g() = log _ () 8. f () = log () and g() = ln() 9. f () = e and g() = ln() For the following eercises, match each function in Figure 9 with the letter corresponding to its graph. B A C Figure 9 0. f () = log ( + ). g() = log ( + ). h() = log ( + ) For the following eercises, sketch the graph of the indicated function.. f () = log ( + ). f () = log(). f () = ln( ) 6. g() = log( + 6) + 7. g() = log(6 ) + 8. h() = _ ln( + ) For the following eercises, write a logarithmic equation corresponding to the graph shown. 9. Use = log () as the parent function. 0. Use f () = log () as the parent function. Download for free at http://cn.org/contents/9b08c9-07f-0-9f8-d6ad9970d@.9

SECTION 6. SECTION EXERCISES. Use f () = log () as the parent function.. Use f () = log () as the parent function. TECHNOLOGY For the following eercises, use a graphing calculator to find approimate solutions to each equation.. log( ) + = ln( ) +. log( ) + = log( ) +. ln( ) = ln( + ) 6. ln( + ) = _ ln( ) + 7. _ log( ) = log( + ) + _ EXTENSIONS 8. Let b be an positive real number such that b. What must log b be equal to? Verif the result. 9. Eplore and discuss the graphs of f () = log _ () and g() = log (). Make a conjecture based on the result. 60. Prove the conjecture made in the previous eercise. 6. What is the domain of the function 6. Use properties of eponents to find the -intercepts of the function f () = log( + + ) algebraicall. Show the steps for solving, and then verif the result b graphing the function. f () = ln _ ( +? Discuss the result. ) Download for free at http://cn.org/contents/9b08c9-07f-0-9f8-d6ad9970d@.9