SAMPLE. Exponential and logarithmic functions

Transcription

1 Objectives C H A P T E R 5 Eponential and logarithmic functions To graph eponential and logarithmic functions. To graph transformations of the graphs of eponential and logarithmic functions. To introduce Euler s number. To revise the inde and logarithm laws. To solve eponential and logarithmic equations. To find rules for the graphs of eponential and logarithmic functions. To find inverses of eponential and logarithmic functions. To appl eponential functions to phsical occurrences of eponential growth and deca. 5. Eponential functions The function, f ) = a,where a R + \{}, iscalled an eponential function or inde function). The graph of f ) = a is shown. The features of the graph of the eponential function with rule f ) = a are: f ) = a f ) = f ) = a The -ais is a horizontal asmptote. As, f ) +. The maimal domain is R. The range of the function is R +. An eponential function is a one-to-one function., a, ), a) 74

2 Chapter 5 Eponential and logarithmic functions 75 Graphing transformations of the graph of f ) = a Translations If the transformation, a translation with mapping, ) + h, + k), is applied to the graph of = a, the image has equation = a h + k. The horizontal asmptote has equation = k. The images of the points with coordinates, ),, ) and, a) are a + h, a ) + k, h, + k) and + h, a + k) respectivel. The range of the image is k, ). Eample Sketch the graph, and state the range, of = + A translation of unit in the positive direction of the -ais and units in the positive direction of the -ais is applied to the graph of = The equation of the asmptote is = The mapping is, ) +, + ), ), 5 ), 4) 5, ), ),, ), ), 4) The range of the function is, ). Reflections If the transformation, a reflection in the -ais determined b the mapping, ), ), is applied to the graph of = a, the image has equation = a. The horizontal asmptote has equation =. The images of the points with coordinates, ),, ) and, a) are a, ),, ) and, a) respectivel. The range of the image is, ). a Eample Sketch the graph of =

3 76 Essential Mathematical Methods &4CAS A reflection in the -ais is applied to the graph of = The mapping is, ), ),, ), ), ), ), ), ), ), ) If the transformation, a reflection in the -ais determined b the mapping, ), ), is applied to the graph of = a, the image has equation = a or = ) ) a or =. a The horizontal asmptote has equation =. The images of the points with coordinates, ),, ) and, a) are, ),, ) and, a) respectivel. The range of the a a image is, ). Eample Sketch the graph of = 6 Dilations A reflection in the -ais is applied to the graph of = 6 The mapping is, ), ), 6), ), ) 6 6, ), ), 6), 6) If the transformation, a dilation of factor k k > ) from the -ais determined b the mapping, ), k), is applied to the graph of = a, the image has equation = ka. The horizontal asmptote has equation =. The images of the points with coordinates, ),, ) and, a) are, k ),, k) and, ka) respectivel. The range of the a a image is, ). Eample 4 Sketch the graph of each of the following: a = 5) b =.)8), ), 6

4 a A dilation of factor from the -ais is applied to the graph of = 5 The mapping is, ), ), ), ) 5 5, ), ), 5), 5) Chapter 5 Eponential and logarithmic functions 77, 5 b A dilation of factor. or ) from the -ais 5 is applied to the graph of = 8 The mapping is, ), 5 ), ), ) 8 4, ), ), 4 5, 8), 8 ) 5, ), 5, 5) If the transformation, a dilation of factor k k >)fromthe -ais determined b the mapping, ) k, ), is applied to the graph of = a, the image has equation = a k. The horizontal asmptote has equation =. The images of the points with coordinates, ),, ) and, a) are k, ),, ) and k, a) respectivel. The range of the a a image is, ). Eample 5 Sketch the graph of each of the following: a = 9 b = a A dilation of factor from the -ais is applied to the graph of = 9 The mapping is, ), ), ), ) 9 9 8, 5, 9), ), ), 9), 9), 9, )

5 78 Essential Mathematical Methods &4CAS b A dilation of factor from the -ais is applied to the graph of = ) The mapping is, ),, ), ), ), ) ), ), Combinations of transformations Eample 6,, ) Sketch the graph, and state the range, of each of the following: a = + b = 4 c = a The transformations, a reflection in the -ais and a translation of units in the positive direction of the -ais, are applied to the graph of = The equation of the asmptote is = The mapping is, ), + ), ), 7 ), ), 4), ), 5) b, 5), 4) 7,, The range of the function is, ). The transformations, a dilation of factor from the -ais followed b a translation of unit in the negative direction of the -ais, are applied to the graph of = 4 The equation of the asmptote is = ) The mapping is, ),,, ) ) 4, 4, ), ) ), 4), The range of the function is, )., 4, )

6 c The transformations, a reflection in the -ais followed b a translation of unit in the positive direction of the -ais and units in the negative, direction of the -ais, are applied to the graph of =. The equation of the asmptote is = The mapping is, ) +, ), ), ), ), ), ), ) The range of the function is, ). Eercise 5A Chapter 5 Eponential and logarithmic functions 79, ), ) For each of the following, use the one set of aes to sketch the graphs and label asmptotes) of: a = and = b = and = c = 5 and = 5 d =.5) and =.5 ) Sketch the graph of each of the following labelling asmptotes), and state the range of each: ) ) a = + b = c = ) ) d = + e = + f = + Sketch the graph of each of the following labelling asmptotes), and state the range of each: a = b = + c = ) ) d = e = + f = 4 For f ) =,sketch the graph of each of the following, labelling asmptotes where appropriate: a = f + ) b = f ) + c = f ) + d = f ) ) e = f ) f = f g = f ) + h = f ) 5 Sketch the graph of each of the following labelling asmptotes), and state the range of each: a = b = + c = d = e = + + f = + 4

7 8 Essential Mathematical Methods &4CAS 6 A bank offers cash loans at.4% interest per da compounded dail. A loan of $ is taken and the interest paable at the end of das is given b C = [.4) ] a Plot the graph of C against. b Find the interest at the end of: i das ii das c After how man das is the interest paable $? d A loan compan offers $ with a charge of $4.5 a da being made. The amount charged after das is given b C = 4.5 i Plot the graph of C against using the same window as in a). ii Find the smallest value of for which C < C. 7 If ou invest $ at % per da, compounding dail, the amount of mone ou would have after das is given b =.) dollars. For how man das would ou have to invest to double our mone? 8 a i Graph =, = and = 5 on the same set of aes. ii Forwhat values of is > > 5? iii Forwhat values of is < < 5? b iv Forwhat values of is ) = = 5? ) ) Repeat part a for =, = and = 5 c Use our answers to parts a and b to sketch the graph of = a for: i a > ii a = iii < a < 5. The eponential function, f ) = e In the previous section the famil of eponential functions f ) = a, a R + \{}, was eplored. One member of this famil is of such importance in mathematics that it is known as the eponential function. This function has the rule f ) = e,where e is Euler s number, named after an eighteenth centur Swiss mathematician. Euler s number is defined as: e = lim + ) n n n To see what the value of e might be, we could tr large values of n and a calculator to evaluate + ) n. n Tr n = then + ) =.) = n =.) = n =.) = n =. ) = n =. ) = As n is taken larger and larger it can be seen that + ) n approaches a limiting value n.78 8). Like, e is irrational: e =

8 Chapter 5 Eponential and logarithmic functions 8 Investigation into the production of glass marbles A method of producing high qualit glass marbles has been proposed. A rack holding small silica cones threaded on a wire will circulate around the track as shown in the diagram. When the rack enters the spra unit it will be subjected to a fine spra of a liquid glass substance. It takes minute to produce a marble. Rack Enclosed spra unit Drive A marble produced b a single passage around the unit will take minute and the volume will be increased b %, i.e. doubled. However, such a large increase in volume, at this slow speed, will tend to produce misshapen marbles. This suggests that the rack should be speeded up. We shall investigate what happens to the volume of the marble as the rack is speeded up and tr to answer the question, Is there a maimum volume reached if the rack speeds up indefinitel? Let V = volume of the marble at time t. Also let the original marble volume equal V. Forpassage per minute V = V. Now assume that if the rack is speeded up to do passages/minute then the growth in volume is 5% for each passage; that is: V = ) ) V = + V =.5V ) and similarl: for 4 passages, V = + 4) 4 V =.44...V for 8 passages, V = + 8) 8 V = V for 6 passages, V = + 6) 6 V =.67...V for 64 passages, V = + 64) 64 V = V for n passages, V = + ) n V n As the rack speeds up, n is taken larger and larger, and it can be seen that + n approaches a limiting value, i.e. V = lim + ) n V n n = ev So the maimum volume of the marble if the rack speeds up indefinitel is ev. ) n

9 8 Essential Mathematical Methods &4CAS Graphing = e The graph of = e is as shown. The graphs of = and = are shown on the same set of aes: Eercise 5B, e,,, ), ), e) Sketch the graph of each of the following and state the range: a f ) = e + b f ) = e c f ) = e d f ) = e e f ) = e f f ) = e g h) = + e ) h h) = e ) i g) = e + j h) = e k f ) = e + l h) = e Solve each of the following equations using a calculator. Give answers correct to three decimal places. a e = + b e = + c = e d = e a Using a calculator plot the graph of = f ) where f ) = e b Using the same screen plot the graphs of: ) i = f ) ii = f iii 5. Eponential equations = f ) In this section the one-to-one propert of eponential functions is eploited to solve eponential equations. This propert can be stated as: Eample 7 a = a implies = Find the value of for which: a 4 = 56 b = 8 = = e = a 4 = 56 4 = 4 4 = 4 b = 8 = 4 = 4 = 5, )

10 Chapter 5 Eponential and logarithmic functions 8 Inde laws The solution of equations ma also require an application of one or more of the inde laws and these are stated here: To multipl two numbers in eponent form with the same base, add the eponents: a m a n = a m+n To divide two numbers in eponent form with the same base, subtract the eponents: a m a n = a m n To raise the power of a to another power, multipl the eponents: Eample 8 a m ) n = a m n a = Find the value of for which: a 5 4 = 5 + b 9 = 7 a 5 4 = 5 + = 5 ) + = = = 8 = Eercise 5C b ) = 7 Let =. = = ) 9) = = or 9 = = or = 9 = or = = or = Simplif the following epressions: a 4 6 b 8 4 c 8 4 d 4 4 ) + ) 4 e 4 ) f 5 5 ) 4 4 ) g ) 4 h 8 6 ) + i + Solve for in each of the following: a = 8 b 8 = 9 c 4 = 56 d 65 = 5 e = 8 f 5 = 5 g 6 = 4 h = i 5 = Solve for n in each of the following: a 5 n 5 n = 65 b 4 n = c 4 n = 56

11 84 Essential Mathematical Methods &4CAS d n 9 = 7 n e n 4 n = 64 f n 4 = 8 4 n g 7 n = 9 n+ h 8 6n+ = 8 4n i 5 4 n = 5 6 n j n 4 n+ = 6 k 7 n ) n = 7 n 4 4 Solve for : a ) + 7 = b ) = c 5 5 ) 5 = d 5 5 ) + 5 = e = 6 ) 8 f 8 ) 6 = ) g ) = 64 h 4 54 ) = 4 i ) = 8 ) 9 j 77 ) = 87 ) 5.4 Logarithmic functions The eponential function f ) = a,where a R + \{}, isaone-to-one function. Therefore, there eists an inverse function see Section.7). To find the rule of the inverse function, do the following. Let = a Therefore = log a Therefore f ) = log a The following definition was used to find the rule of the inverse: log a = if a = Foreample: log 8 = isequivalent to the statement = 8 log. = isequivalent to the statement =. The graphs of = e and its inverse The graphs of = log, = log e function = log e are shown on and = log are shown on the one set the one set of aes. of aes., e =, ) = e, e), ), e = log e e, ), ) = log = log e, ) = log, ) e, ) The features of the graph of the logarithmic function with rule f ) = log a are: f ) = f a) =

12 Chapter 5 Eponential and logarithmic functions 85 The -ais is a vertical asmptote. As +, f ). The maimal domain is R +. The range of the function is R. Alogarithmic function is a one-to-one function. Note: The function with rule f ) = log e is known as the natural logarithm function. Logarithm laws We use the inde laws to establish rules for computations with logarithms. Let a = m and a = n,where m, n and a are positive real numbers. mn = a a = a + log a mn) = + and since = log a m and = log a n it follows that: log a mn) = log a m + log a n Foreample: log + log 5 = log 5) = log ) = m n = a a = a m ) log a = n m ) and so log a = log n a m log a n Foreample: log log 8 = log 8 = log 4 = If m = ) log a = log n a log a n ) = log a n log a = log n a n and so Foreample: m p = a ) p = a p log a m p ) = p log a m p ) = p log a m log 5 = log 5 ) = log 5

13 86 Essential Mathematical Methods &4CAS Eample 9 Without using a calculator, simplif the following: 6 log + log 6 log log + log 6 log 5 = log + log 6 log 5 6 = log 9 + log 6 log 5 = log ) 6 = Logarithmic equations Eample Solve each of the following equations for : a log = 5 b log ) = 4 c log e + ) = a log = 5 = 5 = c log e + ) = + = e = = Using the TI-Nspire Use Solve ) from the Algebra menu b )asshown. ln) = log e ), the logarithm with base e, is available on the kepad b pressing /.Logarithms with other bases are obtained b pressing the log ke / ) and completing the template. b log ) = 4 = 4 = 7 = 7 )

14 Chapter 5 Eponential and logarithmic functions 87 Using the Casio ClassPad Enter and highlight ln ) + ln + ) = ln6 8) then tap Interactive > Equation/inequalit > solve. Ensure the variable is set to. Eample Solve each of the following equations for : a log 7 = b log e ) + log e + ) = log e 6 8) c log log 7 ) = log 6 a log 7 = is equivalent to = 7 B inspection = 9 b log e ) + log e + ) = log e 6 8) log e ) + ) = log e 6 8) + = = ) ) = = or = Note: The solutions must satisf >, + > and 6 8 >, i.e. > 4. Therefore both of these solutions are allowable. c log log 7 ) = log 6 log 7 = log 6 7 = 6 = 4 = 4 = 4

15 88 Essential Mathematical Methods &4CAS Graphing transformations of the graph of f ) = log a Eample Sketch the graph of = log e This is obtained from the graph of = log e badilation of factor from the -ais and a dilation of factor from the -ais. ) The mapping is, ), ), ), ) e, ) e, Eample Sketch the graph, and state the domain, of each of the following: a = log 5) + b = log + 4) a The graph of = log 5) + isobtained from the graph of = log b a translation of 5 units in the positive direction of the -ais and unit in the positive direction of the -ais. The equation of the asmptote is = 5 The mapping is, ) + 5, + ), ) 6, ), ) 7, ) The domain of the function is 5, ). When =, log 5) + = log 5) = 5 = 5 e,, 7, ) 6, ) 5, = 5

16 b Eample 4 Chapter 5 Eponential and logarithmic functions 89 The graph of = log + 4) is obtained from the graph of = log b a translation of 4 units in the negative direction of the -ais and a reflection in the -ais. The equation of the asmptote is = 4 The mapping is, ) 4, ), ), ), ), ) The domain of the function is 4, ). When =, = log + 4) = log 4 Sketch the graph of = log e + 5). The graph of = log e + 5) isobtained from the graph of = log e b adilation of factor from the -ais followed b a translation of 5 units in the negative direction of the -ais and units in the negative direction of the -ais. The equation of the asmptote is = 5 The mapping is, ) 5, ), ) 4, ) e, ) e 5, ) The domain of the function is 5, ). When =, = log e + 5) = log e 5 When =, 4, ) log 4 5 = 5, log e 5 ).58, ) log e + 5) = log e + 5) = and + 5 = e = e 5.58

17 9 Essential Mathematical Methods &4CAS Eercise 5D Evaluate each of the following: a log b log c log 6. d log 64 e log f log 8 Epress the following as the logarithm of a single term: a log e + log e b log e log e 8 c log e + log e + log e d log e + log e 4 e log e + log e 4 + log e f log 5 e uv + log e uv + log e uv g log e + 5log e h log e + ) + log e ) log e ) Solve each of the following equations for : a log = b log = 8 c log e 5) = d log = 6 e log e + 5) = 6 f log e ) = g log e + ) = h log = i log 4) = 4 Solve each of the following equations for : a log = log + log 5 b log e = log e 5 log e c log e = log e 8 d log e + log e ) = e log e log e ) = log e + ) 5 Epress each of the following as the logarithm of a single term: a log 9 + log b log 4 log 6 c e log a log b d + log a log b log 6 log 7 log 64 6 Without using our calculator, evaluate each of the following: a log 5 + log b log 5 + log log 4 c log + log + log d log 5 + log + e 4log log 6 7 Simplif the following epressions: ) a log b log log + log ) c log e ) log e ) log e + ) 8 Solve each of the following equations for : a log e + 8) = log e b log e 5) log e ) =

18 Chapter 5 Eponential and logarithmic functions 9 9 Solve each of the following equations for : a log e ) + log e + ) = b 8e e = Solve for : log e ) + log e 4 = log e 9 ) Given that log a N = log a 4 log a.75 6log a ), find the value of N. Sketch the graphs of each of the following. Label the aes intercepts and asmptotes. State the maimal domain and range of each. a = log e ) b = log e + ) c = log e + ) d = + log e ) e = log e + ) f = log e ) g = log e + ) h = log e ) i + = log e 4 ) Sketch the graphs of each of the following. Label the aes intercepts and asmptotes. State the maimal domain of each. a = log b = log 5) c = log d = log ) e = log 5 ) f = log + g = log h = log 5) + i = 4log ) j = log ) 6 4 Solve each of the following equations using a calculator. Give answers correct to three decimal places. a + = log e b log e + ) = + 5 a Using a calculator plot the graph of = f ) where f ) = log e b Using the same screen plot the graphs of: ) i = f ) ii = f ) iii = f iv = f ) 5.5 Determining rules for graphs of eponential and logarithmic functions In previous chapters, we considered establishing rules for graphs of some functions. In this chapter, we consider similar questions for eponential and logarithmic functions. Eample 5 The rule for the function of the graph is of the form = ae + b. Find the values of a and b.

19 9 Essential Mathematical Methods &4CAS When =, = and when =, = 6 6 = ae + b ) and = ae + b ) Subtract ) from ): 6 = ae e ) 6 = ae ) Eample 6 Therefore a = 6 e.88 From equation ), b = 6 a = 6 6 e = 6e e 5.67 Therefore.88e , 6), ) The rule for the function of the graph shown is of the form = a log e + b). Find the values of a and b. When = 5, = and when = 8, = = a log e 5 + b) ) and = a log e 8 + b) ) From ) log e 5 + b) = 5 + b = e and b = 4 From ) = a log e 4 a = log e log e 4) 5, ) Eample 7 Given that = Ae bt and = 6when t = and = 8when t =, find the values of b and A. 8, )

20 When t =, = 6 Thus 6 = Ae b ) When t =, = 8 Thus 8 = Ae b ) Divide ) b ): 4 = eb 4 b = log e Substitute in ): 6 = Ae log e 4 6 = 4 A A = 8 4 = 9 Hence = 9 elog e 4 )t = 9 ) 4 t ) 9 e.88t Using the TI-Nspire Use Solve ) from the Algebra menu b )asshown. ln) = log e ), the logarithm with base e, is available on the kepad b pressing /.Logarithms with other bases are obtained b pressing the log ke / ) and completing the template. Using the Casio ClassPad Enter and highlight ln ) + ln + ) = ln6 8) then tap Interactive > Equation/inequalit > solve and ensure the variable is set to. Chapter 5 Eponential and logarithmic functions 9

21 94 Essential Mathematical Methods &4CAS Eercise 5E The graph shown has rule: = ae + b Find the values of a and b. The rule for the function for which the graph is shown is of the form: = ae + b Find the values of a and b. The rule for the function f is of the form: f ) = ae + b Find the values of a and b., 6), 7) 4 Find the values of a and b such that the graph of = ae b goes through the points, 5) and 6, ). 5 The rule of the graph shown is of the form: = a log b) Find the values of a and b. = 5 7, ) = 4, 4) = f) = 5 6 Find the values of a and b such that the graph of = ae b goes through the points, ) and 6, 5). 7 Find the values of a and b such that the graph of = a log + b goes through the points 8, ) and, 4). 8 Find the values of a and b such that the graph of = a log b) passes through the points 5, ) and 7, 4).

22 Chapter 5 Eponential and logarithmic functions 95 9 The points, ) and 5, ) lie on the graph of the function with rule = a log e b) + c. The graph has a vertical asmptote with equation =. Find the values of a, b and c. The graph of the function with rule f ) = a log e ) + b passes though the points, 6) and 4, 8). Find the values of a and b. 5.6 Change of base and solution of eponential equations It is often useful to change the base of an eponential or logarithmic function, particularl to base or e since these are the onl ones available on the calculator. To change the base of log a from a to b a > and b > and a, b ), we use the definition that = log a implies a = Taking log b of both sides: log b a = log b Therefore log b a = log b Therefore = log b a log b Since = log a log a = log b a log b or log a = log b log b a This demonstrates that the graph of = log a can = log b, ) b be obtained from the graph of = log b b a dilation of factor from the -ais. log b a A similar process shows that = a can be written as log b a = log b. Rearranging to make the subject: = b log b a) Since = a : a = b log b a) This demonstrates that the graph of = a can be obtained from the graph of = b b a dilation of factor from the -ais. log b a The statement log a = log b can be used log b a to simplif epressions, as in the following eamples., log b a b, ), b, ) b, log b a, b) = log a = b = a, b log b a

23 96 Essential Mathematical Methods &4CAS Eample 8 Simplif: a log e 7 log e log a e 7 log e = log 7 = Eample 9 b log 4 log 4 Evaluate, correct to four significant figures: a log b log a log = log e log e. Eample 6 If log 6 = k log +, find the value of k. log 6 = k log ) + = log k + log = log k ) Therefore 6 = k = k k = Eample b log 4 = log log = 5 b log 6 = log 6 log ).585 Solve for if =, epressing the answer to two decimal places. Take either the log or log e since these are the onl logarithmic functions available on our calculator) of both sides of the equation: Therefore log = log i.e. log = log Therefore = log log.46

24 Chapter 5 Eponential and logarithmic functions 97 Eample Solve = 8, epressing the answer to three decimal places. Eample log e = log e 8 ) log e = log e 8 = log e 8 log e = log e 8 log e + = ) loge 8 log e +.7 Solve { :.7.},epressing the answer to three decimal places. Taking log of both sides: Eercise 5F log.7 log. log.7 log. log. Note the sign change.) log Use our calculator to solve each of the following equations, correct to two decimal places: a = 6 b =.7 c = d 4 = 5 e = 5 f. = g 5 = h 8 = 5 + i = 8 j. + =.7 k = + l.4 + = 5.9) m 5 = n +) = o + = Solve for using a calculator. Epress our answer correct to two decimal places. a < 7 b > 6 c. > d 8 e..4

25 98 Essential Mathematical Methods &4CAS Solve each of the following equations for : a = 5 b = 7 c = 4 Simplif: a log 4 log b log 5 6 log 5 6 c log 4 8 log 4 + log Evaluate, correct to four decimal places: a log 6 b log 57 c log 4 8 ) d log 5 99 e log 7 f log 67 + log a If a log 7 = log 6 4, find the value of a, correct to three significant figures. b If log 8 = log k, find the value of k, correct to one decimal place. 7 Prove that log b a + log c b + log a c = log a b + log b c + log c a 8 Prove that if log r p = q and log q r = p, then log q p = pq 9 If u = log 9, find in terms of u: a b log 9 ) c log 8 Solve the equation log 5 = 6 log 5 Given that q p = 5, find log 5 q in terms of p. 5.7 Inverses It has been observed that f ) = log a and g) = a are inverse functions. In this section this observation is used to find inverses of related functions, and to transform equations. An important consequence is the following: log a a = for all R a log a = for R + Eample 4 Find the inverse of the function f : R R, f ) = e + and state the domain and range of the inverse function. Recall that the transformation a reflection in the line = isgiven b the mapping, ), ). Consider = e + Then = e and = log e ) i.e., the inverse function has rule f ) = log e ) The domain of f = the range of f =, ). The range of f = the domain of f = R.

26 Chapter 5 Eponential and logarithmic functions 99 Eample 5 Rewrite the equation = log e ) + with as the subject. = log e ) + Therefore = log e and = e Eample 6 Find the inverse of the function f :, ) R, f ) = log e ) +. State the domain and range of the inverse. Consider = log e ) + Therefore and Therefore Hence = log e ) = e = e + f ) = e + The domain of f = the range of f = R The range of f =, ). Using the TI-Nspire Use Solve ) from the Algebra menu b ) asshown.

27 Essential Mathematical Methods &4CAS Using the Casio ClassPad Enter and highlight = ln ) + then tap Interactive > Equation/inequalit > solve and ensure the variable is set to. Eample 7 Rewrite the equation P = Ae kt with t as the subject. P = Ae kt Taking logarithms to the base e of both sides: Eercise 5G log e P = log e Ae kt ) log e P = log e A + log e e kt t = k log e P log e A) = ) P k log e A On the one set of aes, sketch the graphs of = f ) and = f ) where f : R R, f ) = e + On the one set of aes, sketch the graphs of = f ) and = f ) where: f :, ) R, f ) = log e ) Find the inverse of each of the following functions and state the domain and range in each case: a f : R + R,where f ) = log e b f : R + R,where f ) = log e ) + c f : R R,where f ) = e + d f : R R,where f ) = e + e f : ), R, where f ) = log e + ) f f : ), R, where f ) = 4log e + ) g f : { : > } R, f ) = log + ) h f : R R, f ) = e

28 Chapter 5 Eponential and logarithmic functions 4 The function f has the rule f ) = e a Sketch the graph of f. b Find the domain of f and f ). c Sketch the graph of f on the same set of aes as the graph of f. 5 Let f : R R, f ) = 5e a Sketch the graph of f. b Find the inverse function f. c Sketch the graph of f on the same set of aes as the graph of f. 6 Let f : R + R, f ) = log e ) + a Sketch the graph of f. b Find the inverse function f and state the range. c Sketch the graph of f on the same set of aes as the graph of f. 7 For each of the formulas, make the pronumeral in brackets the subject: a = log e ) + 5 ) b P = Ae 6 ) c = a n n) d = 5 ) e = 5 log e ) ) f = 6 n n) g = log e ) ) h = 5 e ) ) 8 For f : R R, f ) = e 4: a Find the inverse function f. b Find the coordinates of the points of intersection of the graphs of = f ) and = f ) 9 For f : R R, f ) = log e + ) + 4: a Find the inverse function f. b Find the coordinates of the points of intersection of the graphs of = f ) and = f ) a Using a calculator, for each of the following plot the graphs of = f ) and = g), together with the line =, ontheone set of aes. i f ) = log e and g) = e ii f ) = log e ) + and g) = e iii f ) = log and g) = b Use our answers to part a to comment on the relationship between f ) = a log b + c and g) = b c a

29 Essential Mathematical Methods &4CAS 5.8 Eponential growth and deca Eponential and logarithmic functions are used to model man phsical occurrences. It will be shown in Chapter that if a quantit increases or decreases at a rate which is, at an time, proportional to the quantit present, then the quantit present at time t is given b the law of eponential change. Let A be the quantit at time t. Then, A = A e kt,where A is a constant. Growth: k > Deca: k < The number k is the rate constant of the equation. Phsical situations where this is applicable include: the growth of a cell population growth continuousl compounded interest radioactive deca Newton s law of cooling. Eample 8 A bank pas % interest compounded annuall. You invest $. How does this $ grow as a result of the interest added? End of ear Set out in tabular form. Amount, A $ $. ) % = = + ) $. %) % = + ) + ) = + ) $. %) % % = + ) $. 4 = + ) 4 $464. = + ) $59.74 n = + ) n =.) n $.) n

30 A =.) n, n N {} In general, if P = original investment A = amount the investment growstoafter n ears r = compound interest rate r% per annum n = number of ears invested then A = P + r Eample 9 Chapter 5 Eponential and logarithmic functions A $) ) n n ears) The population of a town was 8 at the beginning of 99 and 5 at the end of 999. Assume that the growth is eponential. a Find the population at the end of. b In what ear will the population be double that of 999? a Let P be the population at time t ears measured from Januar 99). Then P = 8e kt At the end of 999, t = 8 and P = 5. 5 = 8e 8k 5 8 = e8k k = 8 log 5 e 8.79 The rate of increase is 7.9% per annum. Note: The approimation.79 was not used in the calculations which follow. The value for k was held in the calculator. When t = P = 8e k The population is approimatel b When does P =? Consider the equation: = 8e kt 8 = ekt 5 4 = ekt.75 = e kt t = k log e

31 4 Essential Mathematical Methods &4CAS The population reaches approimatel 6.8 ears after the beginning of 99, i.e. during the ear 8. Eercise 5H In the initial period of its life a particular species of tree grows in the manner described b the rule d = d mt where d is the diameter of the tree in centimetres, t ears after the beginning of this period. The diameter after ear is 5 cm and after ears, 8 cm. Calculate the values of the constants d and m. The number of bacteria in a certain culture at time, t weeks, is given b the rule N = N e kt. If when t =, N = and when t = 4, N = calculate the values of N and k. The number of people, N,who have a particular disease at time t ears is given b N = N e kt a b If the number initiall is and the number decreases b % each ear, find: i the value of N ii the value of k How long does it take for 5 people to be infected? 4 Polonium- is a radioactive substance. The deca of polonium- is described b the formula M = M e kt,where M is the mass in grams of polonium- left after t das, and M and k are constants. At time t =, M = g and at t = 4, M = 5g. a Find the values of M and k. b What will be the mass of the polonium- after 7 das? c After how man das is the mass remaining g?

32 Chapter 5 Eponential and logarithmic functions 5 Chapter summar Sketch graphs of = a, e.g. a = orore, and transformations of these graphs: Inde laws a m a n = a m+n a m a n = a m n Logarithms a m ) n = a m n log a = if a =, ) = = e The inverse function of f : R R, f ) = a is f : R + R, f ) = log a Sketch graphs of = log a, e.g. a = or or e, and transformations of these graphs: = log Logarithm laws log a mn) = log a m + log a n m ) log a = log n a m log a n ) log a = log n a n log a m p ) = p log a m, ) = log e = log Change of base log a = log b log b a a = b log b a) Inverse properties = Review log a a = and a log a =

33 6 Essential Mathematical Methods &4CAS Review Law of eponential change Let A be the quantit at time t. Then: A = A e kt, where A is a constant. Growth: k > Deca k < The number k is the rate constant of the equation. Multiple-choice questions If 4 log b =log b 6 + 8,then is equal to: A 4 B ±6 C ±b D 6 E b The epression log e 4e )isequal to: A log e e ) B log e + C log e 4 D log e 4 + E The epression log 4) is equal to: A B 4 C 4) D 4 E log 4 log 4 4 Let f : A R, f ) = e, g: B R, g) =, and h: C R, + h) = e +,where A, B and C are the largest domains for which f, g and h + respectivel are defined. Which one of the following statements is true? A A C and ran g) = ran h) B A = B and ran f ) ran h) C A C and ran f ) = ran h) D B = C and ran g) = ran h) E B = C and ran g) ran h) 5 If = 5isasolution of the equation log k ) =, then the eact value of k is: A B log + C D 5 E log + log 4) is equal to: A 8 B C 4 5 D 8 E log The solution of the equation =. is closest to: A.8 B.8 C D. E.9 8 The graph of the function with equation = ae + b is shown below. The values of a and b respectivel are: A, B, C, D, E,

34 Chapter 5 Eponential and logarithmic functions 7 9 Which one of the following statements is not true of the graph of the function f : R + R, f ) = log 5? A The domain is R +. B The range is R. C It passes through the point 5, ). D It has a vertical asmptote with equation =. E The slope of the tangent at an point on the graph is positive. If log 7log ) = + log, then is equal to: A B C 4 8 ) 4 4 D 4 ) 7 E ) 7 4 Short-answer questions technolog-free) Sketch the graphs of each of the following. Label asmptotes and aes intercepts. a f ) = e b g) = + c h) = e ) d f ) = e e f ) = log e + ) f h) = log e ) + g g) = log e ) h f ) = log e ) a For f : R R, f ) = e, find f. b For f :, ) R, f ) = log e ), find f. c For f :, ) R, f ) = log + ), find f. d For f : R + R, f ) = +, find f. For each of the following, find in terms of : a log e = log e ) + b log = log + c log = log + 4 d log = + 5log e log e = log e f log e = 4 Solve each of the following equations for, epressing our answers in terms of logarithms of base e: a = b =.8 c = + 5 Solve each of the following for : a = b log e ) = c log ) + = d 7 + = 6 The graph of the function with equation = log + ) + intersects the aes at the points a, ) and, b). Find the eact values of a and b. 7 The graph of the function with equation f ) = 5log + ) passes through the point k, 6). Find the value of k. 8 Find the eact value of for which 4e = Find the value of in terms of a for which log a = + log a 8. For the function f :4, ) R, f ) = log 4), state the domain of the inverse function f. The graph of the function with equation f ) = e ke + 5 intersects the aes at, ) and a, )and has a horizontal asmptote at = b. Find the eact values of a, b and k. Review

35 8 Essential Mathematical Methods &4CAS Review Etended-response questions A liquid cools from its original temperature of 9 Ctoatemperature T Cinminutes. Given that T = 9.98), find: a the value of T when = b the value of when T = 7 The population of a village at the beginning of the ear 8 was 4. The population increased so that, after a period of n ears, the new population was 4.6) n.find: a the population at the beginning of 8 b the ear in which the population first reached 5 The value, $V,ofaparticular car can be modelled b the equation: V = ke t where t ears is the age of the car. The car s original price was $ 497, and after ear it is valued at $8. a State the value of k and calculate,giving our answer to two decimal places. b Find the value of the car when it is ears old. 4 The value $M of a particular house in a certain area during the period 988 to 994 can be modelled b the equation M = Ae pt where t is the time in ears after Januar 988. The value of the house on Januar 988 was $65 and its value on Januar 989 was $6. a State the value of A and calculate the value of p, correct to two significant figures. b What was the value of the house in 99? Give our answer to the nearest. 5 There are two species of insects living in a suburb: the Asla bibla and the Cutus pius. The number of Asla bibla alive at time t das after Januar is given b: N A t) = + t t 5 The number of Cutus pius alive at time t das after Januar is given b: N C t) = 8 + t t 5 a With a calculator plot the graphs of = N A t) and = N C t) onthe one screen. b i Find the coordinates of the point of intersection of the two graphs. ii At what time is N C t) = N A t)? iii What is the number of each species of insect at this time? c i Show that N A t) = N C t)ifandonl if t = [ )] + t + log log ii Plot the graphs of = and = [ )] + + log log and find the coordinates of the point of intersection.

36 Chapter 5 Eponential and logarithmic functions 9 d It is found b observation that the model for Cutus pius doesn t quite work. It is known that the model for the population of Asla bibla is satisfactor. The form of the model for Cutus pius is N C t) = 8 + c t. Find the value of c, correct to two decimal places, if it is known that N A 5) = N C 5). 6 The number of a tpe of bacteria is modelled b the formula n = A e Bt )where n is the size of the population at time t hours. A and B are positive constants. a When t =, n = and when t = 4, n = 5. i Show that e 4B e B + =. ii Use the substitution a = e B to show that a a + = iii Solve this equation for a. iv Find the eact value of B. v Find the eact value of A. b Sketch the graph of n against t. c After how man hours is the population of bacteria 8? 7 The barometric pressure, P cm, of mercur at a height h km above sea level is given b P = 75.5h ). Find: a P when h = b P when h = c h when P = 6 8 A radioactive substance is decaing such that the amount A gattime t ears is given b the formula A = A e kt.ifwhent =, A = 6.7 and when t = 6, A = 5, find the values of the constants A and k. 9 In a chemical reaction the amount g) of a substance that has reacted is given b: = 8 e.t ) where t is the time from the beginning of the reaction, in minutes. a Sketch the graph of against t. b Find the amount of substance that has reacted after: i minutes ii minutes iii minutes c Find the time when eactl7gofthesubstance has reacted. Newton s law of cooling for a bod placed in a medium of constant temperature states: T T s = T T s )e kt where: T is the temperature in C) of the bod at time t in minutes) T s is the temperature of the surrounding medium, and T is the initial temperature of the bod. An egg at96 Cisplaced in a sink of water at 5 Ctocool. After 5 minutes the egg s temperature is found to be 4 C. Assume that the temperature of the water does not change.) a Find the value of k. b Find the temperature of the egg when t =. c How long does it take for the egg to reach a temperature of C? Review

37 Essential Mathematical Methods &4CAS Review The population of a colon of small, interesting insects is modelled b the following hbrid function: e.t t 5 Nt) = e 5 t 7 e e 7 t + ) t > 7 where t is the number of das. a Sketch the graph of Nt)against t. b Find: i N) ii N4) iii N6) iv N8) c Find the number of das for the population to reach: i 968 ii 9