Exponential and Logarithmic Functions

Transcription

1 Chapter 7 Eponential and Logarithmic Functions Specific Epectations Identify, through investigations, using graphing calculators or graphing software, the key properties of eponential functions of the form a (a 0, a ) and their graphs. Describe the graphical implications of changes in the parameters a, b, and c in the equation y ca b. Determine the limit of an eponential function. Determine, from the equation of an eponential function, the key features of the graph of the function, using the techniques of differential calculus, and sketch the graph by hand. Define the logarithmic function log a (a ) as the inverse of the eponential function a, and compare the properties of the two functions. Epress logarithmic equations in eponential form, and vice versa. Simplify and evaluate epressions containing logarithms. Solve eponential and logarithmic equations, using the laws of logarithms. Solve simple problems involving logarithmic scales. Pose and solve problems related to models of eponential functions drawn from a variety of applications, and communicate the solutions with clarity and justification. Identify e as lim n n Define ln as the inverse function of e. n and approimate the limit, using informal methods. Determine the derivatives of combinations of the basic polynomial, rational, eponential, and logarithmic functions, using the rules for sums, differences, products, quotients, and compositions of functions. Determine the equation of the tangent to the graph of an eponential or a logarithmic function. Determine, from the equation of a simple combination of polynomial, rational, or eponential functions, the key features of the graph of the combination of functions, using the techniques of differential calculus, and sketch the graph by hand. Determine the derivatives of the eponential functions a and e and the logarithmic functions log a and ln. Section , , 7.6, 7.7, , , , 7.7

2 Specific Epectations Describe the significance of eponential growth or decay within the contet of applications represented by various mathematical models. Compare the rates of change of the graphs of eponential and non-eponential functions. Solve problems of rates of change drawn from a variety of applications involving eponential or logarithmic functions. Section 7.6, 7.7, MODELLING M AT H What is the source of power for satellites that orbit Earth? Will that power ever run out? Why do mountain climbers need oygen at high altitudes? Why do airplane cabins need to be pressurized? How can we compare the strength of earthquakes, and relate it to how much damage is done? You will investigate these and other questions in this chapter. All of the phenomena mentioned here involve changes that can be modelled by eponential functions. You will see that eponential models predict increases that are more and more rapid, or decreases that approach certain values asymptotically. Such models are widely applicable to many situations in nature.

3 Review of Prerequisite Skills. Eponent laws Simplify. a) (5) b) c) 0 d) + e) 5 f). Writing numbers in eponential form Write each number as a power with the indicated base. a) 6, base b), base c), base d), base 6. Eponent laws Simplify, using the eponent laws. a) b) c) 6 ( 7y) y ( fg ) h fg h + y ( + y) 6 d) m n 5 mn e) ( 9cd) ( 7st ) f) (a b ) (a b). Transformations Describe the transformations performed on the graph of y f(). a) y f () b) y f () c) y f () d) y f () e) y f () f) y 0.5f () 7 g) h) y = f + y 7f () ( 7) Transformations Graph each function using transformations on the graph of y. a) y b) y ( ) c) y d) y e) y ( 5) 7 6. Compound interest Use the formula for compound interest, A P( i) n, to find the amount of interest earned and the final amount for each investment. a) $00 invested at 5%, compounded annually, for 8 years b) $7000 invested at 7.5%, compounded annually, for.5 years c) $000 invested at.5%, compounded semi-annually, for years d) $900 invested at 7.5%, compounded semi-annually, for 5 years e) $0000 invested at %, compounded quarterly, for years f) $600 invested at.75%, compounded monthly, for 7 years 7. Inverses a) Find the inverse of each function. i) y ii) y iii) y b) Decide whether each inverse in part a) is a function. Eplain. 8. Limits (Sections 6., 6.5) Determine each limit. a) lim b) lim 0 0 c) lim d) lim 0 e) 5 lim f) lim 9. Asymptotes (Sections 6., 6.5) Sketch a graph having the following properties. a) vertical asymptote at 0, horizontal asymptote at y 0, function is increasing for 0 and increasing for 0 b) vertical asymptote at, horizontal asymptote at y, function is concave up c) vertical asymptote at, oblique asymptote y, function is concave down d) vertical asymptotes at and, horizontal asymptotes at y and y MHR Chapter 7

4 0. Polynomial equations (Section.5) Solve for. a) 0 0 b) c) d) 6 0. Eponential equations William has $00 to spend. He spends half of his money on the first day, half of the remaining amount on the second day, half of the remaining amount on the third day, and so on. (The amount he spends each day is rounded up to the nearest cent.) How long does his money last?. Eponential equations Gwen performs one push-up this week, two the net week, and doubles the amount every week thereafter. After how many weeks is she performing more than 000 push-ups?. Eponential equations Dilip and Sara start with no money. Dilip receives $0 the first day, $0 the second day, $0 the third day, and so on, with the amounts increasing by $0 each day. Sara receives $ the first day, $ the second day, $ the third day, and so on, with the amounts doubling each day. a) Who receives at least $50 on a single day first? b) Who receives at least $00 on a single day first? c) Sketch a graph of the amount received on each day for both Dilip and Sara for the first 0 days.. Eponential equations You win a prize and are offered Option A, which is $ , or Option B, which is $0.0 on the first day, $0.0 on the second day, and the amount doubles each subsequent day. a) Determine the accumulated amount for Option B after days, days, days, and 5 days. b) After how many days is the accumulated amount in Option B greater than the amount received in Option A? c) Repeat part b) if Option A is to receive $ d) Repeat part b) if Option A is to receive $ e) Repeat part b) if Option A is to receive $ f) Repeat part b) if Option A is to receive $ per day. Historical Bite: What is a Mathematician? The mathematician Paul Halmos (96 ) eplains what a mathematician does: The mathematician is interested in etreme cases in this respect he is like the industrial eperimenter who breaks lightbulbs, tears sheets, and bounces cars on ruts. How widely does reasoning apply, he wants to know, and what happens when it doesn t? What happens when you weaken one of the assumptions, or under what conditions can you strengthen one of the conclusions? It is this perpetual asking of such questions that makes for a broader understanding, better technique, and greater elasticity for future problems. Review of Prerequisite Skills MHR

5 7. Eponential Functions Eponential functions, such as y and y =, are used to model a wide variety of natural phenomena. For eample, a bacterial culture that doubles in size every hour might be modelled by the function y t, where t is in hours. The amount of a radioactive isotope t with a half-life of 000 years might be modelled by the function y =, where t is in 000 years. Eponential functions can be written in the form f() a, where the base, a, is a positive constant not equal to, that is, a (0, ) or a (, ). The eponent,, is a variable that can be any real number, unless there is some restriction on the domain. Eponential relationships are used to model compound interest, population growth, and resource consumption. We can perform all the usual transformations translations, reflections, and stretches on the graphs of eponential functions. The following investigation eplores the effect of these transformations on eponential functions. Investigate & Inquire: Transforming Eponential Functions. Graph the function y. Then, graph each function below and use the information from the graphs to copy and complete the table. Equation y Transformation of graph of y Domain Range End Behaviour Equation of asymptote(s) Intercepts y 5 y () y () y (). a) Describe the effect of c on the graph when transforming y a into y ca. What happens to the graph if c 0? b) Describe the effect of b on the graph when transforming y a into y a b i) if b 0 ii) if b 0. Without constructing a table of values, a) describe how to graph y (), given the graph of y b) describe how to graph y ()5 7, given the graph of y 5 c) describe how to graph y ba c, given the graph of y a Net, we will eplore changing the value of the base, a. MHR Chapter 7

6 Investigate & Inquire: How the Base of an Eponential Function Influences Its Graph. Use graphing technology to investigate the graphs of eponential functions with different bases. a) Graph y, y, y 5, and y. b) Graph,,, and y =. y = y = y = 5. a) Describe the changes to the graphs of y a as a increases, for a (, ). b) Describe the changes to the graphs of y a as a decreases, for a (0, ).. a) Compare the graphs in step, parts a) and b), in pairs. That is, compare the graphs of y and, and then the graphs of y and y =, and so on. How are y = the graphs in each pair alike? How are they different? b) Without graphing, describe how the graphs of y 6 and y = differ and what they 6 have in common. Use graphing technology to confirm your answer.. What point do all eponential functions appear to have in common? Eplain why this is so. 5. a) Graph the functions y 0 and y. Describe the graphs. Are these eponential functions? Eplain. b) What happens if you try to graph y () on a graphing calculator or graphing software? Set up a table of values of y () using the TABLE SETUP screen, with a TblStart value of and a Tbl value of 0.. Eplain why we have the restriction a (0, ) or a (, ) for the eponential function y a. The graph of f () a, where a (, ) and (, ), is continuous and always increasing. When a (0, ), the graph is continuous and always decreasing. Look at the graphs of the eponential functions f() a for various values of the base a. Notice that, regardless of the base, all of these graphs pass through the same point, (0, ), because a 0 for a 0. Also, note that the -ais is a horizontal asymptote and that the graphs never touch or cross the -ais, since a 0 for all values of. Thus, the eponential function f () a has domain (, ) and range (0, ) for a 0. Now, we take a closer look at the function y a, where a (0, ) or a (, ). a (, ): As approaches infinity, the graph of y a increases rapidly, and as approaches negative infinity, the graph is asymptotic to the -ais. Thus: If a (, ), lim a 0 and lim a. 6 y = ( 5 y y = ( 5 6 ( ( y = ( ( y = ( ( y = y = 7. Eponential Functions MHR 5

7 Furthermore, the larger the base a, the more rapidly the function increases as approaches infinity. As approaches negative infinity, the larger the base a, the more rapidly the graph of the function approaches its asymptote, the -ais. y y =5 8 y = ( 5 ( 8 y 6 y = 6 y = ( ( (0, ) 0 (0, ) 0 a (0, ): As approaches infinity, the graph is asymptotic to the -ais, and as approaches negative infinity, the graph of y a increases rapidly. Thus: If a (0, ), lim a and lim a 0. Furthermore, the smaller the base a, the more rapidly the graph of the function approaches its asymptote, the -ais, as approaches infinity. As approaches negative infinity, the smaller the base a, the more rapidly the function increases. Can a be less than 0 for y a? That is, can y a have a negative base? The answer is yes, but the resulting function is so badly discontinuous that it has no practical use. In the Investigation above, y () was graphed. Note from the table of values that, for many values of, the function is not defined. For eample, ( ), or, is not 6 defined in the real numbers. Neither are ( ), ( ), and so on. Since many epressions with negative bases cannot be evaluated, we restrict the definition of the eponential function to positive values of a. Since for all values of, f() is not considered an eponential function. Since 0 0, > 0 = undefined, 0 f () 0 is not considered an eponential function either. Thus, we restrict a in the eponential function y a to positive real numbers not equal to, that is, a (0, ) or a (, ). Eample Transformations of an Eponential Function Use the graph of y to sketch the graph of each function. Use technology to confirm your results. a) y b) y (Note: this is not the same as y (), which is a discontinuous function.) 6 MHR Chapter 7

8 Solution a) The graph of y is obtained by starting with the graph of y and translating it four units upward. We see from the graph that the line y is a horizontal asymptote. The asymptote has moved up units from the original asymptote, y 0. y y y = y = y = 6 y = b) Again, we start with the graph of y and reflect it in the -ais to get the graph of y. The horizontal asymptote is y 0. Eample Transformations of an Eponential Function a) Use the graph of y to sketch the graph of y. b) State the asymptote, the domain, and the range of this function. Solution a) To graph y using the graph of y, we translate the original graph units to the right. Then, we graph y by translating y downward units. To summarize, we graph y by translating the graph of y to the right units and downward units. b) We see from the graph that the horizontal asymptote is y. It has been translated downward units along with the graph. The domain is R and the range is (, ). y y = 6 ( ) y = Eample An Eponential Model for Light Transmitted by Water The equation s 0.8 d models the fraction of sunlight, s, that reaches a scuba diver under water, where d is the depth of the diver, in metres. a) Use graphing technology to graph the sunlight, s 0.8 d, that reaches a scuba diver, using a suitable domain and range. b) Determine what percent of sunlight reaches a diver m below the surface of the water. 7. Eponential Functions MHR 7

9 Solution a) The graph of s 0.8 d is shown. For Window variables, we use the domain [0, 0] because depth cannot be negative. We use trial and error to find a suitable upper limit of the domain. Since s is a proportion, we use the range [0, ]. b) Use the key to find the value of s when d. Confirm the solution algebraically. s d This means that only about 50% of the sunlight above the water will reach a diver m below the surface of the water. Eample Limit of an Eponential Function Find lim. Solution Graphing Calculator Method Since the function f( )= is not continuous at, we cannot substitute to find the limit. We will use a table of values to determine the limit. Enter the function in the Y= editor of a graphing calculator. Then, set up a table using the TABLE SETUP screen. Use Ask mode for the independent variable (, in this case). Enter -values that approach from the left. From the table, it appears that approaches zero very quickly as approaches from the left. Thus, it appears that = lim 0. Solution Paper and Pencil Method As approaches from the left, approaches zero, and is negative. Thus, approaches negative infinity. We can rewrite lim z as lim, where z =. Recall z = from earlier that, if a (, ), then lim z a 0. Thus, lim 0, so lim 0. z 8 MHR Chapter 7

10 Key Concepts If a (, ), then lim a 0 and lim a. The function f () a has domain (, ) and range y (0, ). If a (0, ), then lim a and lim a 0. The function f () a has domain (, ) and range (0, ). The graph of y a is a reflection of the graph of y a in the -ais. The graph of y ca p b is obtained by graphing the original function y a, and transforming it as follows: a) Stretch the graph vertically by a factor of c if c ; compress the graph vertically by a factor of c if 0 c. b) If c 0, reflect the graph in the -ais. c) Translate the graph left p units if p 0, and right p units if p 0. d) Translate the graph up b units if b 0, and down b units if b 0. Communicate Your Understanding. a) Is the domain of every eponential function the same? Eplain. b) Is the range of every eponential function the same? Eplain.. a) Given the graph of y, eplain how to graph y without using technology or a table of values. b) Given the graph of y 6, describe how to graph y ()6 5.. Eplain why the function y a is not considered an eponential function a) when a b) when a 0 c) when a 0 Practise A. a) Eplain the similarities and differences among the graphs of y, y 6, and y 9. In your eplanation, pay attention to the y-intercepts and the limits as approaches positive and negative infinity. b) Repeat part a) for the graphs of y,, and y. y 6 9. Graph each pair of functions on the same set of aes. First, use a table of values to graph f (), and then, use your graph of f () to graph g(). Check your work using graphing technology. a) f () and g ( ) b) f () 5 and g ( ) 5 c) f () 6 and g ( ) 6 d) f () 0 and g ( ) 0. a) Use a table of values to graph f (). b) Describe how to graph g ( ) ( ) and h() () using the graph of f (). c) Graph g ( ) ( ) and h() () on the same set of aes as f (). Confirm your results with graphing technology. d) State the domain and range of each function. 7. Eponential Functions MHR 9

11 . Use the given graphs to sketch a graph of each of the following functions without using a table of values or technology. State the y-intercept, domain, range, and equation of the asymptote of each function. y a) f () b) g() c) h() ()7 5 d) e) f( )= f) g ( )= + h ( )= y 6 y = 6. Evaluate. a) lim b) c) lim d) e) lim f) 0 g) lim h) 6 i) lim ( ) j) lim 8 lim lim 0 + lim 5 lim 0 y y = y = a) f () b) g() c) h() d) f () () 5 e) f) f( )= y = + g) h) 5. h ( ) = ( ) y = () + For each function, i) state the y-intercept ii) state the domain and range iii) state the equation of the asymptote iv) draw a graph of the function B Apply, Solve, Communicate 7. Application In Eample on page 7, it would be almost completely dark underwater if only 0.% of the sunlight above the water reached a diver. At what depth does this occur? 8. Inquiry/Problem Solving Randolf bought a computer system for $000. The system depreciates at a rate of 0% each year. a) Determine an eponential function to model the value of the system over time. b) Graph the function you found in part a). c) Use your graph to determine the value of Randolf s computer system years from now. d) Verify your result from part c) algebraically. 9. Inquiry/Problem Solving Roy bought an antique slide rule in 00 for $5. The value of the collector s item is increasing and can be approimately determined by the epression y 5(.0) t, where t is the number of years since 00, and y is the value, in dollars. a) Graph the value of the slide rule over time. 0 MHR Chapter 7

12 b) Find the approimate value of the slide rule in 05. c) Determine approimately when the slide rule will have a value of $ Collette bought a $500 compound interest savings bond with a 5% annual interest rate. a) Graph the growth of Collette s savings bond over time. b) Use your graph to determine the value of her bond when it matures 5 years from now. Web Connection To find current and historical values for the Bank of Canada prime lending rate, go to and follow the link.. Application If f (), show that h f( + h) f( ) =. h h. Communication A sample of radioactive iodine- atoms has a half-life of about 8 days. This means that after 8 days, half of the atoms will have transformed into some other type of atom. A formula that models the number of t iodine- atoms that remain is 8 P P0 ( ), where P is the number of iodine- atoms that remain after time t, in days, and P 0 is the number of iodine- atoms that are initially present. Suppose that iodine- atoms are initially present. a) How many iodine- atoms remain after days? b) How many iodine- atoms remain after 80 days? c) How many iodine- atoms remain after 60 days? C d) Does the result in part c) make sense? Comment on the domain of validity of the model. e) According to nuclear physics, the transformation of radioactive atoms is a discrete process, that is, every so often an individual atom transforms. There is always a whole number of untransformed atoms remaining. Thus, a graph of this process would not be continuous. Yet the eponential model is continuous. Comment on the validity of using a continuous model for a phenomenon that is essentially discrete. Eplain why the model is nevertheless a good one. Over what domain is the model good?. Eplain how you would graph the following functions given the graph of y. Then, graph each function and state its domain and range. a) y b) y. Evaluate each limit. a) lim b) 5 c) lim d) lim 5 lim 5. In this section, we stated that eponential functions with positive bases are continuous functions. In this eercise, you will eplore the meaning of eponential epressions with irrational eponents. 5 a) Eplain the meaning of in terms of powers and roots. b) Eplain the meaning of 5. in terms of powers and roots. c) Eplain the meaning of 5. in terms of powers and roots. d) Eplain the meaning of 5. in terms of powers and roots. e) Eplain the meaning of 5. in terms of powers and roots. f) Use the idea behind parts a) to e), and the idea of the limit, to eplain the meaning of Eponential Functions MHR

13 7. Logarithmic Functions Suppose you have an investment and would like to determine how long it will take to double. If you start with $500 invested at 6% per year, compounded annually, the value, A, in dollars, of your investment after t years is given by A 500(.06) t. To find out how long the investment will take to double, you need to solve the equation (.06) t, or (.06) t. This equation can be solved by using trial and error on a scientific calculator, by using a graphical approimation, or by using logarithms. Logarithmic functions are related to eponential functions in a special way. In the following investigation, you will discover how they are related. Investigate & Inquire: Inverse of an Eponential Function. a) The number of bacteria in a culture starts at, and doubles every hour. Let y represent the number of bacteria, and represent the time, in hours. Make a table of values relating and y. b) What eponential function relates y to?. To determine when there will be 5 bacteria in the culture, what equation would have to be solved?. a) Graph y and its inverse on the same set of aes. Recall that, to find the inverse function, interchange the - and y-values. b) Describe the graphical relationship between a function and its inverse, and their relationship to the line y. c) Eplain how the graph of the inverse can be used to approimate the solution to the equation determined in step.. Find an approimate solution for each equation by carefully graphing an appropriate function and its inverse. a) b) 6 7 c) 9 5. Make a general statement eplaining how to solve eponential equations graphically. To find the inverse of the eponential function y a, we echange and y to obtain a y. The resulting inverse function is called a logarithmic function. If a y, then the logarithm of to base a, y log a, is defined as the eponent to which the base a must be raised to obtain. For eample, if we want to evaluate log 9, the question is, To what eponent must be raised to give 9? The answer is, so log 9. In eponential form, log 9 is written 9. y = log is equivalent to = a a eponent base base eponent y MHR Chapter 7

14 Eample Evaluating Logarithms Evaluate. a) log b) log 8 c) Solution log 6 a) We can evaluate log by determining to what eponent the base must be raised to get the result. Since 5, the eponent is 5. Thus, log 5. b) We can evaluate log 8 by determining to what eponent the base must be raised to get the result 8. Since 8, the eponent is. Thus, log 8. c) We can evaluate log by determining to what eponent the base must be raised to 6 get the result. Since is between zero and one, the eponent must be negative. We 6 6 know that =. Thus, log =. 6 6 The results of Eample can be summarized in the following table. Logarithmic Form Eponential Form log 5 5 log 8 8 log 6 = = 6 Since the functions y a and y log a are inverses of each other, their graphs are reflections of each other in the line y. Note also the intercepts. For y a, we see that a 0, so the y-intercept is. Since a 0 can also be written as log a 0, we see that the -intercept of y log a is. From Section 7., we know that the domain of y a is the set of all real numbers, or (, ), and the range is y (0, ). In general, for inverses, the domain and range are reversed. That is, the domain of f () is equal to the range of f (), and the range of f () is equal to the domain of f (). So, the domain of y log a is (0, ) and the range is y (, ). 6 (, 0) Look carefully at the domain of the function y log a and note that 0. Another way of looking at this is by rewriting y log a as a y. Since a y is always positive, 0. Since a (0, ) or a (, ) for the function y a, a (0, ) or a (, ) for its inverse y log a. In this section, we will be looking only at values of a in the interval (, ). (0, ) 7. Logarithmic Functions MHR y 6 0 y = a y = y= log a

15 Since the graph of y a has a horizontal asymptote (the -ais), its inverse, y log a, has a vertical asymptote (the y-ais). We can also see this by looking at the ranges of the two functions. Observing the behaviour of the graphs, we note that lim loga. It is more difficult 0 to tell the behaviour of y log a as from the graph. Is there a horizontal asymptote? If y log a had a horizontal asymptote, then its inverse, y a, would have a vertical asymptote. Since y a does not have a vertical asymptote, y log a does not have a horizontal asymptote. Thus, lim log, if a. Properties of Logarithms If we rewrite the logarithmic function y log a as a y, we can discover other properties of logarithms. a Investigate & Inquire: Eploring Logarithms. Rewrite each logarithm as an eponential, and then determine y. a) y log b) y log 5 c) y log 7 d) y log 00. Make and test a conjecture about the value of log a, for any a.. Repeat step for the following logarithms. a) y log b) y log 8 8 c) y log d) y log 0 0. Make and test a conjecture about the value of log a a, for any a. 5. Repeat step for the following logarithms. a) y log ( ) b) y log 7 (7 ) c) y log ( ) d) y log 9 (9 ) 6. Make and test a conjecture about the value of log a a, for any a and any R. log 7. The equation y = can be rewritten using logarithms. First, let z log. Then, y z z log y log log y Rewrite each power as a logarithm, and then determine y. log log log a) b) c) y 8 y 7 7 y 7 d) log y 85 log 8. Make and test a conjecture about the value of a a, if a and Make a list of your conjectures from steps,, 6, and 8. These are four very important properties of logarithms. MHR Chapter 7

16 Eample Transformations of Logarithmic Functions Graph each function and its inverse on the same set of aes. a) y log b) y log c) y log () Solution a) The inverse of y log is y. Graph y log. Then, reflect the graph in the line y to obtain the graph of its inverse, y. y 8 6 y = y = y= log b) To graph y log, we start with the graph of y log from part a), and reflect it in the -ais. To find the inverse of y log, switch the - and y-coordinates and solve for y. log y log y y To graph y, reflect the graph of y log in the line y. y 6 y = 0 y= log y = 6 c) To graph y log (), we start with the graph of y log from part a), and then, reflect it in the y-ais. y= log ( ) y y = To find the inverse of y log (), reverse the - and y-coordinates. log (y) y y 6 0 y = To graph y, reflect the graph of y log () in the line y. 7. Logarithmic Functions MHR 5

17 Eample Graphing Logarithmic Functions Graph each function and state the domain, the range, and the equation of the vertical asymptote. a) y log 0 ( ) b) f () log Solution a) We obtain the graph of y log 0 ( ) by translating the graph of y log 0 to the right unit. From the graph, we observe that the domain is (, ), the range is y (, ), and the vertical asymptote is the line. y y y=+log y= log 0 y= log y= log ( ) 0 b) The graph of f () log is obtained by translating the graph of f () log upward units. From the graph, we observe that the domain is (0, ), the range is y (, ), and the vertical asymptote is the y-ais. Eample Determining the Doubling Time for an Investment Solve the equation (.06) t to determine how long it will take to double $500 invested at 6%, compounded annually. Solution We use graphing technology to graph y 500(.06). We are trying to determine the -value for which y 000. So, we graph y 000 on the same screen, and find the point of intersection. Window variables: [0, 5], y [0, 00] We use the Intersect operation to determine that y 000 when.90. Therefore, the investment doubles in about years. 6 MHR Chapter 7

18 In Eample, we solved the equation (.06), by graphing. This equation can be rewritten as (.06), or log.06. Unfortunately, the key on a calculator determines the logarithm only to base 0, so we cannot use it (yet) to evaluate a logarithm to base.06. In Section 7., we will obtain a formula for changing the base of a logarithm. With the help of this formula and a calculator, logarithms to any base can be determined. For now, we will solve equations such as the one in Eample by graphing. Eample 5 Domain of a Logarithmic Function Without graphing, find the domain of the function f () log (9 ). Solution The function f () log a is defined only when 0. This means that the function f () log (9 ) is defined only when (9 ) 0. (9 ) 0 ( )( ) 0 Use an interval chart to determine when this inequality is true. Interval Test value Sign of ( )( + ) ( ) ( + ) (, ) (, ) (, ) The product is positive when (, ). Thus, the domain of f () log (9 ) is (, ). Key Concepts The inverse of y a is y log a. y log a is equivalent to a y. The function y log a has domain (0, ) and range y (, ). Four important properties of logarithms are log a 0 log a a log a (a ) log a a The key on a calculator determines the logarithm to base 0. Communicate Your Understanding. Compare and contrast the graphs of y 5 and y log 5.. a) Eplain how to obtain the graph of y log a given the graph of y a. b) Eplain the relationship between eponential functions and logarithmic functions.. Given y log a, eplain what happens if the base is.. Compare and contrast the domain, range, intercepts, and asymptotes for eponential and logarithmic functions. 7. Logarithmic Functions MHR 7

19 A Practise. Copy and complete the table. Logarithmic Form. Evaluate. a) log b) log c) log 9 d) log 6 e) log 7 f) log 6 log log 6 6 log 9 log 5 = 5 log 7 0 log = g) h) Eponential Form log Use graphing technology to graph each pair of functions on the same set of aes. State the domain and range of each function. a) y and y log b) y and y log c) f () 8 and g() log 8. Determine the equation of the inverse of each function. Graph each equation and its inverse. a) y b) y log 5. Graph each function using transformations. State the domain, the range, and the equation of the vertical asymptote. a) f () log b) y log () c) f () log 0 d) f () log ( ) e) y log 5 ( ) f) y log ( ) = 9 = = 6 B 6. Without graphing, determine the domain of each function. a) f () log (5 ) b) y log 7 (7 ) c) y log ( ) d) y log 0 ( ) Apply, Solve, Communicate 7. How long will it take to triple an investment of $000 at 7.5%, compounded annually? 8. Communication a) Write an equation to represent the accumulated amount of a $500 mutual fund invested at 8%, compounded annually. Determine how long it will take for the investment to i) increase to $000 ii) double b) If the interest rate is doubled, how is the doubling time of the investment affected? Eplain. 9. Application Wyatt invested $500 at 8.5%, compounded annually, when he was 8 years old. Use a graph to determine Wyatt s age when his original investment has doubled. 0. Communication a) The number of gophers living in a field in the summer can be modelled by the equation y 00(.) n, where n is the number of years from the present. Plot the graph of the gopher population to determine how long it will take for the gopher population to i) increase to 50 ii) double b) i) Using the equation in part a), determine the number of gophers after 000 years. Is this realistic? ii) What do you think is the domain of validity of the equation in part a)? Eplain. iii) What might affect the gopher population so that the equation in part a) is no longer valid?. Application The intensity of light in a particular river is reduced by % for each metre below the surface of the water. This relationship can be modelled by the equation I(d) I 0 (0.96) d, where I(d) is the intensity of light, in lumens, at depth d, in metres, and I 0 is the original intensity. Use a graph to determine how far 8 MHR Chapter 7

20 below the water s surface the light has to travel so that the intensity is a) 0.8I 0 b) 0.5I 0 c) 0.I 0. Inquiry/Problem Solving Pollution affects the clarity of water. The intensity of light below a particular polluted river s surface is reduced by 5% for each metre below the surface of the water. Write an eponential equation to model this relationship, using question as a guide. a) What percent of the original intensity of light penetrates to m below the surface of the water? b) At what depth does 0% of the original intensity of light remain? c) What implications does the reduction of the sun s intensity have for plant life under water?. How long will it take, to the nearest month, for $500 to grow to $000, if it is invested at 7%, compounded monthly?. During the 990s, the world s population was growing at a rate of.% per year. In 999, the world s population reached 6 billion. a) Assuming the growth rate remains constant, write an equation relating the population to the time in years after 999. b) Sketch a graph of the relation in part a). c) Use the graph to determine the world s population in 05. d) Use the graph to predict the year in which the world s population will have doubled since Inquiry/Problem Solving The formula K [[log 0 (n)]] is useful in computer programming. (The function [[]] truncates by removing the decimal part; for eample, C [[]], [[.79]].) Apply this formula with a few positive integers (both large and small) and state a hypothesis about what the formula determines. 6. There are initially 000 bacteria in a culture. The number of bacteria doubles every hour, so the number of bacteria after t hours will be N 000() t. a) Plot a graph of this relation. b) When does the formula cease to be valid? 7. The ph of a chemical solution is a measure of its acidity and is defined as ph log 0 [H ], where H is the concentration of hydrogen ions in moles per litre. a) Graph the relation. b) What is the ph value of a solution with a hydrogen concentration of mol/l? c) Find the hydrogen concentration for a ph value of a) Find the domain of the function f () log (log 0 ). b) Find f (). 9. In this section, we have discussed only logarithms with positive integer bases, such as,,, and so on. Can logarithms have other bases? Graph each function and its inverse. Is the inverse a continuous function? a) f( )= b) y (0.5) c) g ( )= 7. Logarithmic Functions MHR 9

21 7. Laws of Logarithms With the properties of logarithms learned in Section 7., we can work with only a limited number of logarithmic situations. There are other properties of logarithms that are useful for solving eponential and logarithmic equations. Such equations arise in a variety of contets, such as investments and bacterial growth. In the following investigation, you will eplore three very important properties of logarithms, dealing with products, powers, and quotients. Investigate & Inquire: Laws of Logarithms. a) Copy and complete the table. log a log a y log log 8 log a (y) log log 9 log 7 log log 6 log log 5 log 6 log 6 log 0 log 5 5 log 5 5 log 5 5 b) Eamine the results of each row. Make a conjecture about the product law for logarithms. c) Test your conjecture by evaluating log 6 (6 6), and make any necessary adjustments to your original conjecture.. Make and test a conjecture about the power law of logarithms: log a p c c log a p.. a) Copy and complete the table. log a log a y log a y log log 8 log 8 = log 7 log 9 log 7 9 = log log 6 log 6 = log 6 log 6 log 6 6 = log 5 5 log 5 5 log = b) Eamine the results of each row. Make a conjecture about the quotient law for logarithms. c) Test your conjecture by evaluating log, and make any necessary adjustments to 7 your original conjecture. 0 MHR Chapter 7

22 The patterns in the investigation show three laws of logarithms. Product law: log a (pq) log a p log a q Power law: log a (p c ) clog a p p Quotient law: log a q log a p log a q Because logarithms can be written as eponents, the laws of eponents can be used to justify corresponding laws of logarithms. To show the product law, we let log a p X and log a q Y. Then, rewriting in eponential form, log a p X becomes a X p and log a q Y becomes a Y q So, log a (pq) log a (a X a Y ) (substitution) log a (a XY ) (eponents law) X Y (log a a ) Substituting for X and Y, log a (pq) log a p log a q. Web Connection For a visual eplanation of the product law of logarithms, go to and follow the link. In questions and on page 5, you will show how to derive the power law for logarithms and the quotient law for logarithms. Since logarithms with base 0 are very common, log 0 is usually written as log. As we mentioned in Section 7., the key on a calculator determines logarithms to base 0. Eample The Laws of Logarithms Evaluate each epression using the laws of logarithms. a) log 6 log 6 9 b) log 9 c) log log d) log log 50 e) log log 7 f) log 9 Solution a) log 6 log 6 9 log 6 ( 9) (product law) log 6 6 log 6 (6 ) c) log log log log 8 b) log 9 log 9 ( ) (power law) log 9 9 This epression can be evaluated without using the power law. log 9 log 9 9 d) Remember, log means log 0. log log 50 log ()(50) (product law) log 00 log 0 7. Laws of Logarithms MHR

23 e) log log 7 log log 7 6 (quotient law) f) log 9 log ( 9 ) log 9 (power law) Eample Using the Laws of Logarithms Epress as a single logarithm. a) log 7 0 log 7 0 b) log + log 7 log c) log ( ) log ( ) Solution a) log 0 log 0 log log b) Remember, log means log 0. log log 7log log log 7 log (quotient law) (power law) log ( )( 7 ) log 6 7 (product and quotient laws) c) log ( ) log ( ) log ( ) + = + log ( + )( = ) + = log ( ), > Eample Using the Laws of Logarithms Epand. a) log 6 ( y ) b) Solution log a bc a) log 6 ( y ) log 6 log 6 y log 6 log 6 y a b) log = log a log ( bc) bc = log a (log b+ log c) = log a log b log c Recall Eample in Section 7. (page 6), which we could solve only by graphing because the calculator key determines logarithms only to base 0. With a simple formula, we can change the base of the logarithm, and use the calculator to evaluate it. For eample, MHR Chapter 7

24 suppose we want to evaluate log on a calculator. To change the base to 0, we start with y log and proceed as follows. y log Write in eponential form. y Take the logarithm to base 0 of each side of the equation. log 0 ( y ) log 0 Use the power law. y log 0 log 0 Solve for y. y = log 0 log0 log0 Thus, log =. log0 We can now evaluate the logarithm on a calculator. The general formula for converting a logarithm from one base to another, called the loga change of base formula, is logb. log b Eample Using the Change of Base Formula a Write each logarithm with base 0, and then evaluate it on a calculator. Round your results to four decimal places. a) log 5 b) log 7 Solution log0 a) log5 = b) log log log 7 log Eample 5 Doubling Time for an Investment How long does it take for an investment of $500 to double at 7% interest, compounded annually? Solution The amount, A, in dollars, of an investment of $500 at 7% interest, compounded annually, for t years, is A 500(.07) t. For the investment to double, A $000. Thus, (.07) t (.07) t t log.07 Use the change of base formula to change to base 0. t = log log It takes approimately 0 years and months for an investment of $500 to double at 7% interest, compounded annually. 7. Laws of Logarithms MHR

25 Key Concepts For a (, ), p, q (0, ), c (, ), a) Product Law: log a (pq) log a p log a q b) p Quotient Law: loga loga ploga q q c) Power Law: log a (p c ) c log a p. log means log 0. To change the base of a logarithm from b to a, use the change of base formula loga logb =. log b a Communicate Your Understanding. Is it possible to use the quotient law of logarithms to evaluate log 7? Eplain.. Is it possible to use the product law of logarithms to evaluate log 7 log 8? Eplain.. Eplain why the base must be changed to evaluate log 5 using a calculator.. Is there more than one way to evaluate log 9 log? Eplain. 5. Is log 5 equal to log 5? Eplain. A Practise. Copy and complete the table. Single Logarithm log ( 5) log 6 (kg) h log f Sum or Difference of Logarithms log log log 8 log 8 log log 5 log 0 log 0 7 log 6 log log 5 log. Rewrite each epression using the power law. a) log b) log c) log d) log e) log 6 f) log 9. Epress as a single logarithm. a) log 5 log 8 log 5 b) log 8 log 0 log c) log 9 log log d) e) log 7 log 5 log (a b) log (a ) f) log ( y) log ( y) g) log log y h) log ab log bc. Rewrite each epression with no logarithms of products, quotients, or powers. a) log 7 (5) b) log (m n ) c) log (abc) d) m log ( y + y ) e) log 8 n f) log 6 (y) 5 g) ab log h) log c jk 5. Evaluate. a) log 8 log 8 b) log 7 log 9 c) log 9 log d) log 9 log 6 e) log 6 log 8 log f) log 08 log g) log 8 6 log 8 log 8 h) log 80 log 5 i) log.5 log 80 j) log MHR Chapter 7

26 B 6. Evaluate to four decimal places using a calculator. a) log 5 b) log c) log 7 9 d) log 6 5 e) log 9 8 f) log 6 g) log 7 h) log 8 Apply, Solve, Communicate 7. Application Driving in fog at night greatly reduces the intensity of light from an approaching car. The relationship between the distance, d, in metres, that your car is from the approaching car and the intensity of light, I(d), in lumens (lm), at distance d, is given by Id d log ( ) 5 a) Solve the equation for I(d). b) How far away from you is an approaching car if I(d) 0 lm? 8. Inquiry/Problem Solving Energy is needed to transport a substance from outside a living cell to inside the cell. This energy is measured in kilocalories per gram molecule, and is given by C the relationship E. log, where C C represents the concentration of the substance outside the cell, and C represents the concentration inside the cell. a) Find the energy needed to transport the eterior substance into the cell if the concentration of the substance inside the cell is i) double the concentration outside the cell ii) triple the concentration outside the cell b) What is the sign of E if C C? Eplain what this means in terms of the cell. 9. Communication Which is greater, log 6 7 or log 8 9? Eplain. 0. The formula for the gain in voltage of an electronic device is A v 0(log V o log V i ), where V o is the output voltage and V i is the input voltage. a) Rewrite the formula as a single logarithm. b) Verify the gain in voltage for V o.8 and V i using both versions of the formula.. When a rope is wrapped around a fied circular object, the relationship between the C larger tension T L and the smaller tension T S is TL modelled by 0. log, where is the TS friction coefficient and is the wrap angle in radians. T L a) Rewrite the formula using the laws of logarithms. b) If the wrap angle is (in radians), and a 00 N force is balancing a 50 N force, what is the friction coefficient? c) If the rope is wrapped around the object.5 times, what force is now needed to balance the 50 N force?. Show that if log b a c and log y b c, then log a y c.. Use the product law of logarithms to prove the quotient law of logarithms, p loga loga ploga q q where a (, ), p, q (0, ).. Use the product and quotient laws of logarithms to prove the power law of logarithms log a (p c ) c log a p, where a (, ), p, q (0, ), c (, ). 5. Derive the change of base formula, logb log. a = log a b 6. Find the error in the following calculation. log 0. log 0. log (0.) log 0.0 log 0. log 0.0 Thus, T S 7. Laws of Logarithms MHR 5

27 7. Eponential and Logarithmic Equations There are many applications of eponential and logarithmic equations, including problems involving atmospheric pressure, the intensity of light passing through glass or water, and the power source for satellites. In order to solve eponential equations that model such applications, we need to solve for the variable in the eponent. We do this by rewriting the equation in terms of logarithms, usually to base 0 to make using a calculator easier. (Recall that log means log 0.) Consider Eample. Eample An Eponential Equation Solve the equation. Solution Paper and Pencil Method log log (base 0 logarithm of each side) log log (power law) log log. 87 We can check the solution by substituting.87 into the original equation..87 Solution Graphing Calculator Method We can also find the solution to graphically, as we did in Section 7.. First, we input y and y in the Y= editor of a graphing calculator or graphing software. Then, we use the Intersect operation to find the -coordinate of the point of intersection of the two graphs. The solution is.87, to four decimal places. Window variables: [.7,.7], y [.,.] 6 MHR Chapter 7

28 Eample Solving Eponential Equations Using Logarithms Solve a) 5 0 b) 6 7 c) (7 ) 8 Solution a) 5 0 log 5 log 0 (base 0 logarithm of each side) ( ) log 5 log 0 (power law) log 0 log 5 log 0 log 5 log 0 log 5 0. b) 6 7 log 6 log 7 ( 6) log log 7 log 7 = log c) (7 ) 8 7 log 7 log ( ) log 7 log log log Here are guidelines for solving eponential equations similar to those in Eample.. Isolate the term containing the variable on one side of the equation.. Take the base 0 logarithm of each side of the equation.. Apply the power law of logarithms to rewrite the equation without eponents.. Solve for the variable and check the result. We will use these steps to solve the problem in Eample. Eample Satellite Power Supply MODELLING M AT H The power source used by satellites is called a radioisotope. The power output of the radioisotope is given by the equation P 50(0.996 t ), where P is the power, in watts, and t is the time, in years. If the equipment in the satellite needs at least 5 W of power to function, for how long can the satellite operate before needing recharging? 7. Eponential and Logarithmic Equations MHR 7

29 Solution We need to determine the value of t when P 5. P 50(0.996 t ) 5 50(0.996 t ) 5 t = t log 0. log t log 0. t log t = log 0. log Thus, in theory, the satellite can operate for about 00 years. We can also solve logarithmic equations using the laws of logarithms. Eample A Logarithmic Equation Solve the equation log ( ). Solution We can solve this equation algebraically using the laws of logarithms. Using the property y log a a y, we can rewrite the equation log ( ) in eponential form, and then solve. log ( ) ( ) 6 The solution is. Eample 5 Solving Logarithmic Equations Solve and check. a) log 5 b) log 9 ( 5) log 9 ( ) c) log log 7 Solution a) log Check 00000: L.S. log R.S. 5 log log L.S. R.S. The solution is MHR Chapter 7

30 b) log 9 ( 5) log 9 ( ) log 9 [( 5)( )] log 9 ( 5) ( 6)( ) 0 6 or Check 6: L.S. log 9 ( 5) log 9 ( ) R.S. log 9 (6 5) log 9 (6 ) log 9 log L.S. R.S. Check : L.S. log 9 ( 5) log 9 ( ) R.S. log 9 ( 5) log 9 ( ) log 9 (9) log 9 () c) log log 7 log log 7 7 Check : L.S. log R.S. log 7 log log 7 log 7 L.S. R.S. The root is. This cannot be evaluated, because the logarithm of a negative number is undefined. Thus, is an etraneous solution. The solution is 6. Here are guidelines for solving logarithmic equations such as those in Eamples and 5.. Isolate the terms with variables to one side of the equation.. Use the laws of logarithms to epress each side of the equation as a single logarithm.. Simplify each side of the equation.. Solve and check. Eample 6 The Relationship Between Altitude and Atmospheric Pressure MODELLING M AT H Atmospheric pressure, P, depends on the altitude above sea level, and is measured in kilopascals (kpa). For altitudes up to 0 km above sea level, the atmospheric pressure is approimately P 0.(.), where is the altitude, in kilometres. A mountain climber is eperiencing atmospheric pressure of 89 kpa. How high above sea level is the mountain climber, to the nearest 0 m? 7. Eponential and Logarithmic Equations MHR 9

31 Solution P 0.(.) 89 0.(.) log = log log 0. = log. 07. The mountain climber is 00 m above sea level, to the nearest 0 m. Net, we will solve a more comple eponential equation. Eample 7 Solving an Eponential Equation by Factoring Solve 0, and illustrate your results graphically. Solution First, we factor to isolate the eponential term. If we let z, it becomes clearer that the equation is quadratic and can be factored. z z 0 (z )(z ) 0 Substituting z back into the equation, we get ( )( ) 0 0 or 0 log log Since 0, there is no solution for. log log = log log. 69 The solution is.69. We can also represent the solution graphically, using the Zero operation. Window variables: [, ], y [0, 50] 0 MHR Chapter 7

32 Key Concepts To solve eponential equations of the form found in Eamples and, first isolate the term containing the eponential variable on one side of the equation, then take the logarithm of each side of the equation, and apply the laws of logarithms to solve for the variable. To solve logarithmic equations of the form found in Eamples and 5, first isolate the terms with variables on one side of the equation, then use the laws of logarithms to epress each side of the equation as a single logarithm, and simplify to solve for the variable. Communicate Your Understanding. Describe two ways to verify the solution(s) to a logarithmic equation or an eponential equation.. Eplain why logarithms are helpful in solving eponential equations.. Give an eample of an eponential equation that cannot be solved eactly using the laws of logarithms. How would you solve the equation in this case? A Practise Round your solutions to four decimal places, if necessary.. Determine if 0.6 is a root of 5, and justify the result.. Solve for and check your solution. a) log 0 b) 7 7 c) 9 d) log 8 e) log 9 f) log 0 g) log 8 = h) log a) Solve each equation using the properties of logarithms. b) Illustrate each solution graphically. i) log ( 6) ii) log 6 ( ). Solve. a) 5 8 b) 6 0 c) = 7 d) 5 e) 6 8 f) 9 g) 6() 8 h) (7) Solve and check. a) log 5 log 5 65 b) log log 7 log c) log6 n = log6 6 d) log log log e) log log 8 log f) log log 5 g) log 6 log 6 5 h) log 7 log Solve for. Use a graphing calculator to verify your solution. a) log b) log ( ) 0 c) log ( ) 9 d) log (5 ) e) log ( 6) log log 0 f) log log ( ) log () g) log 6 ( ) log 6 ( ) h) log ( ) log ( ) i) log ( ) log ( ) j) log ( ) log ( 5) 7. Communication a) Show that log 5 is a root of the equation b) Are there any other roots? Eplain. 8. Solve for and check your solution. a) 6 0 b) ( ) 5 0 c) 7 (7) 0 0 d) 0 5(0) 0 e) 6 (6) 5 0 f) 9() 0 7. Eponential and Logarithmic Equations MHR

33 B Apply, Solve, Communicate 9. The function A P(.06) n represents the amount, A, in dollars, in an investment n years from now, where P is the original investment, or principal, in dollars. If Seth invests $000 now, find how long it will take to accumulate to a) $6. b) $66.08 c) double his original investment d) $ Application The intensity of light, I, in lumens, passing through a certain type of glass is given by I(t) I 0 (0.97), where I 0 is the initial intensity and is the thickness of the glass, in centimetres. a) What thickness of the glass will reduce the intensity of light to half its initial value? b) What effect does doubling the thickness of the glass have on the intensity of light passing through it?. Application The average annual salary, S, in dollars, of employees at a particular job in a manufacturing company is modelled by the equation S 5000(.05) n, where $5000 is the initial salary, which increases at 5% per year. a) How long will it take the salary to increase by 50%? b) If the starting salary is $5 000, how long will it take the salary to increase by 50%? Eplain your answer.. How long, to the nearest month, will it take for an investment of $600 at 5.5%, compounded annually, to a) double? b) triple? c) accumulate to $900?. Inquiry/Problem Solving The speed, v, in kilometres per hour, of a water skier who drops the towrope, can be given by the formula v v 0 (0) 0.t, where v 0 is the skier s speed at the time she drops the rope, and t is the time, in seconds, after she drops the rope. If the skier drops the rope when travelling at a speed of 65 km/h, how long will it take her to slow to a speed of km/h?. Inquiry/Problem Solving On average, the number of items, N, per day, on an assembly line, that a quality assurance trainee can inspect is N 0 (0.7) t, where t is the number of days worked. a) After how many days of training will the employee be able to inspect items? b) The company epects an eperienced quality assurance employee to inspect 5 items per day. After the training period of 5 days is complete, how close will the trainee be to the eperienced employee s quota? 5. The number of hours, H(t), that cheese will remain safe to eat decreases eponentially as the temperature of the surrounding air, t, in degrees Celsius, increases. For a particular type of cheese, this relationship is represented by H(t) 0(0) 0.0t. To the nearest hour, how long will the cheese remain safe to eat if it is stored at a) 0C? b) 6C? c) 5C? 6. Use the formula from Eample 6 (page 9) to determine the altitude of a rock climber, if the atmospheric pressure is approimately 95 kpa. 7. The designers of aircraft must know the eternal pressure in order to control the pressure inside the aircraft. What range of eternal pressure must be controlled for a small airplane with a maimum altitude of 0 km? 8. The intensity of the sunlight below the surface of a large body of water is reduced by.6% for every metre below the surface. Show that the depth at which the sunlight has intensity I(d), in lumens, is given by Id d 8. 9 log ( ), where I 0 is the initial I0 intensity and d is the depth, in metres. 9. Solve for and check your solution. a) log log log 8 log 6 5 b) (5 6 ) 9(5 ) (5 ) 6 0 MODELLING MODELLING M AT H M AT H MHR Chapter 7

34 Logarithmic Scales 7.5 Logarithmic scales are useful for measuring quantities that can have a very large range, because logarithms enable us to make large or small numbers more manageable to work with. Eamples of logarithmic scales include the Richter scale, which measures earthquakes, the ph scale, which measures acidity, and the decibel scale, which measures sound. The intensities of earthquakes vary over an etremely wide range. To make such a wide range more manageable, a compressed range, called the Richter scale, is used. To make the scale convenient, a standard earthquake, with a certain intensity I 0, is given a magnitude of 0. Earthquakes with intensities weaker than this standard are so weak that they are hardly ever discussed. Only magnitudes greater than 0 are used in practice. I The magnitude, M, of an earthquake is given by the equation M = log, where I is the I0 intensity of the earthquake, and I 0 is the intensity of a standard earthquake. Thus, a range of earthquake intensities from I 0 to about I 0 corresponds to a range in magnitudes on the Richter scale from 0 to about 8.9. I The equation M = log can also be written I I 0 0 M. From this equation, you can see I0 that, for every increase in the intensity of an earthquake by a factor of 0, the magnitude on the Richter scale increases by. For eample, an earthquake of magnitude is 0 times as intense as an earthquake of magnitude, and 00 times as intense as an earthquake of magnitude. Eample The Richter Scale MODELLING M AT H a) On September 6, 00, an earthquake in North Bay measured 5.0 on the Richter scale. What is the magnitude of an earthquake times as intense as North Bay s earthquake? b) On February 0, 000, Welland eperienced an earthquake of magnitude. on the Richter scale. On July, 00, St. Catharines eperienced an earthquake of magnitude.. How many times more intense was the earthquake in Welland? 7.5 Logarithmic Scales MHR

35 Solution I a) We use the formula M = log, and let I be the intensity of North Bay s earthquake. I0 I Thus, log = 50.. I0 An earthquake three times as intense as North Bay s earthquake has an intensity of I. So, the magnitude of an earthquake three times as intense as North Bay s earthquake is I M log I0 I log I0 I log log I0 log 5. 0 Substitute the magnitude of North Bay s earthquake So, an earthquake of magnitude 5.5 is three times as intense as an earthquake of magnitude 5.0. b) Let M., the magnitude of the earthquake in Welland, and let M., I the magnitude of the earthquake in St. Catharines. We want to find. We use I the eponential form of the Richter scale equation, I I 0 0 M. I I M I0 0 M I0 0 M M Therefore, the earthquake in Welland was almost 6 times as intense as the earthquake in St. Catharines. The ph scale, which measures the acidity of substances, is another logarithmic scale. The ph of a solution is a measure of relative acidity in moles per litre, mol/l, compared with neutral water, which has a ph of 7. If ph 7, the solution is classified as acidic, and if ph 7, the solution is basic or alkaline. The ph scale ranges from 0 to. The ph scale is widely used by chemists, for eample, who regularly test the ph level of drinking water to ensure that it is safe from contaminants. The ph of a solution can be represented by the equation ph = log, where [H ] is the number of moles of hydrogen [H + ] ions per litre. This can be rewritten as follows to eliminate the fraction. ph log [H ] log[h ] log[h ] (power law) MHR Chapter 7

36 The ph scale makes very small numbers manageable. For eample, if [H ] mol/l, ph 6. The equation ph log [H ] can be rewritten as [H ] 0 ph. Thus, when the ph level increases by, [H ] is divided by 0. For eample, a substance with ph has the H concentration of a substance with ph, and the H concentration of 0 00 a substance with ph. Eample Liquids and ph a) The hydrogen concentration of a sample of water is moles of H per litre of water. What is the ph level of the water? b) A sample of orange juice has a ph level of.5. Find its hydrogen ion concentration. Solution a) To solve for ph, we use the formula ph log [H ] and substitute [H ] ph log [H ] log ( ) 7.66 Therefore, the water has a ph level of approimately 7., which means it is slightly basic. b) We use the formula ph log [H ] and solve for [H ] to determine the hydrogen ion concentration of the juice. ph log [H ].5 log [H ].5 log [H ] [H ] Thus, the hydrogen ion concentration of the orange juice is about. 0 mol/l. The decibel scale, db, also a logarithmic scale, measures sound levels. The human ear can detect a very wide range of sounds, ranging from a soft whisper to loud machinery. The threshold of pain is about 0 db. Eamples of other decibel measurements include normal conversation at about 50 db, and a jet takeoff at about 0 db. The decibel is one tenth of a bel, which is named after the inventor Aleander Graham Bell. Bell is most famous for having invented the telephone, but he worked on many other projects throughout his life, including developing hydrofoils and teaching people with hearing disabilities to speak. The minimum intensity detectable by the human ear is I 0 0 W/m (watts per square metre), and is used as the reference point. The sound level corresponding to an I intensity I watts per square metre is L = 0 log. I Eample Sound Levels and Risk of Hearing Damage Damage to the ear can occur with sound levels that are greater than or equal to 85 db. Find the sound level of a rock concert with an intensity of 80 W/m to determine if fans at the concert are at risk for hearing damage Logarithmic Scales MHR 5

37 Solution We use the decibel formula, and substitute I 80 and I 0 0. I L = 0 log I 0 80 = 0 log Therefore, the concert has a sound level of 9 db, which means that people attending the concert may be at risk for hearing damage. Key Concepts Logarithmic scales convert large ranges of numbers into smaller, more manageable ranges of numbers. Some applications of logarithmic scales are the Richter scale, which applies to earthquakes; the ph scale, which applies to acidity levels; and the decibel scale, which applies to sound levels. Communicate Your Understanding. Can an earthquake have a negative magnitude on the Richter scale? If so, what kind of an earthquake would it be?. Eplain how to determine how many times as intense an earthquake of magnitude 5 is as an earthquake of magnitude.. a) Describe what ph measures. b) Which would you epect to have a higher ph level, baking soda or vinegar? Eplain.. If the pain threshold for sound is 0 db, and a jet engine has a sound intensity level of 0 db, what does this imply for an air traffic controller? a maintenance worker on the ground? A MODELLING M AT H Apply, Solve, Communicate. In 995, Japan had an earthquake of magnitude 7.. What is the magnitude of an earthquake that is a) twice as intense? b).5 times as intense? c) times as intense?. Application Human blood must be maintained in a ph range of 7.5 to 7.5. Calculate the corresponding [H ] range.. a) Show that, if one sound is 0 decibels louder than a second sound, then the first sound is 0 times as intense as the second sound. b) A hair dryer has a sound intensity level of 70 db and an air conditioner has a sound intensity level of 50 db. How many times as intense is the sound from the hair dryer as the sound from the air conditioner?. Inquiry/Problem Solving Earthquakes of magnitude 7.0 or greater can cause metal buildings to collapse. On December, 985, in Mackenzie MODELLING M AT H 6 MHR Chapter 7