Exponential and logarithmic. functions. Topic: Exponential and logarithmic functions and applications
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1 MQ Mths B Yr Ch 07 Pge 7 Mondy, October 9, 00 7: AM 7 Eponentil nd logrithmic functions syllbus ref efer erence ence Topic: Eponentil nd logrithmic functions nd pplictions In this ch chpter pter 7A Inde lws 7B Negtive nd rtionl powers 7C Indicil equtions 7D Grphs of eponentil functions 7E Logrithms 7F Solving logrithmic equtions 7G Applictions of eponentil nd logrithmic functions
2 8 Mths Quest Mths B Yer for Queenslnd Introduction There re one million millimetres in kilometre. It would tke person lmost two weeks of non-stop counting, dy nd night, to count to one million. Although it is difficult to comprehend numbers of this size, t times we must do so. Consider, for emple, mesuring the intensity of sound rnging from quiet room to jet engine. How cn we epress the loudness of these sounds without using numbers so lrge or so smll tht the mind cnnot comprehend them? The nswer involves the use of scle bsed on eponents or powers of number. Eponents or, more specificlly, eponentil functions provide powerful mens of describing phenomen such s rdioctive decy, popultion growth or the vlue of investments. Inde lws Recll tht number,, which is multiplied by itself n times cn be represented in inde nottion. n n lots of where is the bse number nd n is the inde (or power or eponent). n is red s to the power of n or to the n. Multipliction When multiplying two numbers in inde form with the sme bse, dd the indices. m n m + n For emple, 7 Division When dividing two numbers in inde form with the sme bse, subtrct the indices. m n m n For emple, 6 Rising to power To rise n indicil epression to power, multiply the indices. For emple, ( ) + + Inde (or power or eponent) Bse ( m ) n m n mn
3 Chpter 7 Eponentil nd logrithmic functions 9 Rising to the power of zero Any number rised to the power of zero is equl to one. 0, 0 For emple 0 [] or ( ) ( ) 8 8 So [] Using [] nd [] we hve 0. Products nd quotients Note the following. For emple, ( ) ( ) ( ) ( ) ( ) (b) n n b n b n - n b n Simplify ech of the following. y y b ( y ) y c () b 6 9 b d Collect plin numbers ( nd ) nd terms with the sme bse. Simplify by multiplying plin numbers nd dding powers with the sme bse. y y y y 8 y 8 p 6 m ( p) m 6 p m b Remove the brcket by multiplying the powers. (The power of the inside the brcket is.) Convert to plin number () first nd collect terms with the sme bse. Simplify by dding powers with the sme bse. b ( y ) y y 6 y y 6 y y 0 c Write the quotient s frction. c () b 6 9 b ( ) b 6 9 b Remove the brcket by multiplying the powers. b b Continued over pge
4 60 Mths Quest Mths B Yer for Queenslnd Simplify by first cncelling plin numbers. Complete simplifiction by subtrcting powers with the sme bse. (Note:.) 7 b 6 b 7b d Epnd the brckets by rising ech term to the power of. Convert to 7 nd collect like pronumerls. Simplify by first reducing the plin numbers, nd the pronumerls by dding the indices for multipliction nd subtrcting the indices for division. 8 p d 6 m ( p) m 8 p - 6 m p m 6 p m 6 p m Simplify the indices of ech bse. 6p m p 6 p m m - 6 p m 6p 6 + m + 6 b Simplify b 6 b. b 6 Write the epression. Chnge the division sign to multipliction nd replce the second term with its reciprocl (turn the second term upside down). Remove the brckets by multiplying the powers. Collect plin numbers nd terms with the sme bse. 6 b 6 7 b 6 Cncel plin numbers nd pply inde lws. 0 b 0 9 Simplify. b - b 6 b 6 7 b 6 b - b 6 b 9 b b 6 6 b b Epressions involving just numbers nd numericl indices cn be simplified using inde lws nd then evluted.
5 Chpter 7 Eponentil nd logrithmic functions 6 Write in simplest inde nottion nd evlute. 9 6 b 7 Rewrite the bses in terms of their 6 ( ) prime fctors. Simplify the brckets using inde ( ) nottion. Remove the brckets by multiplying the 8 powers. Simplify by dding the powers. Evlute s bsic number. 08 b 9 Rewrite the bses in terms of their b ( ) prime fctors. 7 ( ) ( Simplify the brckets using inde ) - nottion. ( ) Remove the brckets by multiplying the powers. 9 Write in simplest inde form. Evlute s bsic number. Comple epressions involving terms with different bses hve to be simplified by replcing ech bse with its prime fctors. n 8 n + Simplify -. 6 n Rewrite the bses in terms of their 8 n ( ) n prime fctors. 6 n ( ) Simplify the brckets using inde ( ) n nottion. ( ) Remove the brckets by multiplying n n powers. n n Collect terms with the sme bse by dding the powers in the products nd subtrcting the powers in the quotients. n n n Simplify. 6n n n n n n + + n + ( n ) + + n + n +
6 6 Mths Quest Mths B Yer for Queenslnd remember remember. Inde lws: () m n m + n (b) m n m n (c) 0 (d) ( m ) n mn (e) (b) n n b n (f) b n n b n. To simplify indicil epressions: () when deling with questions in the form (epression ) (epression ), replce epression with its reciprocl nd chnge to (b) remove brckets using lws (d), (e) nd (f) (c) collect plin numbers nd terms of the sme bse (d) simplify using lws (), (b) nd (c). 7A Inde lws Mthcd Indices, b c Simplify ech of the following. b m m p p c 7 ( ) d y y y 7 e (y) y f ( ) ( ) g m p (mp ) m p 6 h y (y ) ( y) Simplify ech of the following. 7 b 8 b b b 9 () b c ( )y 6 y d p q 0 (pq ) e (mn ) n f g r s 0 t r (s ) (t) h i Simplify ech of the following. d 6 p 8 m p 7 m 6 9 p m c e y 0 y u v 9 ( u ) v u 6 v 6w t 7 9w t ( w) t b d f b b 6 b 7 b ( ) y 6 y 0 7 y ( e ) f 8e f 0ef ( ) y ( 7 y) - 8 ( y)
7 Chpter 7 Eponentil nd logrithmic functions 6 g ( y ) - y 6 7 y 6 ( y) h ( mp) m p - ( mp) i m p ( mp ) ( mp ) j ( u 7 v 6 ) - ( u v ) u ( v ) Simplify ech of the following. 8 b b 9 b b - b k d - ( k ) 6kd ( k d ) c g ( p ) - g p 8g p 7 ( gp) d jn ( j n) n n ( j) e y 7 y y y f 6 y 8 ( y - ) ( y ) 8 y 7 The frction b The product c The quotient p m - p m cn be simplified to: A p m B p m 6 C p m 8 D p m E 6 6 y - y ( y) cn be simplified to: A y B y C 0 y 6 D 9 y - 0 E b - b b is equl to: - A 8 B 6 C 6 b D - E 6 b 6 6 Simplify ech of the following. n + y z n n y n z n b ( n y m + ) n + y m y n y m 7 Write in simplest inde nottion. 8 b c d 0 8 e f Write in simplest inde nottion nd evlute. b c d 7 9 ( 6 7 ) ( ) ( 6) e ( ) - f - g 8 h ( ) ( ) ( ) 7 8-9
8 6 Mths Quest Mths B Yer for Queenslnd 9 Simplify ech of the following. n 9 - n + 6 n b n n n + c - 6 d n 7 n 6 n 9 n 8 n e n 7 n 9n + n + f n n g 6 n 9 - h * + 8 n 6 n n + - n + n + n i * n - n n n *Hint: Fctorise the numertor nd denomintor first. 0 6 In simplest inde nottion, n 6 n + is equl to: 6 n A 6 n + B 6 n + C 6 n + D 6 9 E 6 n + 9 Negtive nd rtionl powers Negtive powers Wherever possible, negtive inde numbers should be epressed with positive inde numbers using the simple rule: When n inde number is moved from the numertor to denomintor or vice vers, the sign of the power chnges. This is esily verified s follows: n 0 n n n since 0 In other words, n - nd - Note: Chnge the level, chnge the sign. -, 0 0 n using division rule for indices n simplifying the inde. n n n
9 Chpter 7 Eponentil nd logrithmic functions 6 Epress ech of the following with positive inde numbers. 8 y b ( y ) y b Remove the brckets by rising the denomintor nd numertor to the power of. Interchnge the numertor nd denomintor, chnging the signs of the powers. Simplify by epressing s frction to the power of. Remove the brckets by multiplying powers. Collect terms with the sme bse by dding the powers on the numertor nd subtrcting the powers on the denomintor. Rewrite the nswer with positive powers b y ( y) y 0 y y 6 y 7 y 6 y ( ) y 7 y 0 y 0 Rtionl powers Until now, the indices hve ll been integers. In theory, n inde cn be ny number. We will confine ourselves to the cse of indices which re rtionl numbers (frctions). n, where n is positive integer, is defined s the nth root of. n n For emple, we know tht but Therefore,. Similrly,,... etc. n is defined for ll 0 if n is positive integer. In generl, for ny rtionl number, m n - + ( n) m ( n n ) m m
10 66 Mths Quest Mths B Yer for Queenslnd Evlute ech of the following without clcultor. 6 b 9-6 b Rewrite the bse number in terms of its 6 ( ) prime fctors. Remove the brckets by multiplying the 6 powers. Evlute s bsic number. 6 9 Rewrite the bse numbers of the b - frction in terms of their prime fctors. Remove the brckets by multiplying the powers. Rewrite with positive powers by interchnging the numertor nd denomintor. Evlute the numertor nd denomintor s bsic numbers. 7 Simplify the following, epressing your nswer with positive indices. 6 b y 6 y Write the epression. 6 Write using frctionl indices. 6 Write 6 nd in inde form. ( ) ( ) Multiply the powers. Simplify the powers. 9 7 b Write the epression. b y 6 y Epress the roots in inde nottion. ( y 6 ) ( y ) Remove the brckets by multiplying the powers. y y 6 y 6 y 6 7 Collect terms with the sme bse by subtrcting the powers. Simplify the powers. Rewrite with positive powers. - 6 y 9
11 Chpter 7 Eponentil nd logrithmic functions 67 The Mths Quest CD-Rom contins Mthcd file which cn be used to evlute numbers rised to negtive or rtionl powers. A smple screen is shown t right. Mthcd Negtive nd rtionl powers remember remember n, 0 n m - n n n n ( ) m ( n ) m n m 7B Negtive nd rtionl powers Epress ech of the following with positive inde numbers. 6 b c ( ) 7 d ( ) e ( ) 9 f ( 6 ) g ( ) h Simplify ech of the following, epressing your nswer with positive inde numbers. b ( ) ( b ) ( y ) - c ( y ) d ( ) ( e ) ( ) - f ( ) ( ) ( ) Evlute the following without clcultor. 6 9 b c d 7 6 ( m) m - ( p ) p y ( y ) ( ) ( y ) 6 8 SkillSHEET Negtive nd rtionl powers 7. Mthcd e 8 f 8 g 8 h i - j - k - l m n - o p
12 68 Mths Quest Mths B Yer for Queenslnd The ect vlue of 6 is: A B 6 C D - E b The ect vlue of 7-8 is: A B 6 C D E c simplifies to: A B C D E Simplify ech of the following, epressing your nswer with positive indices. 9 8 b 6 c 9 8 d ( ) e ( y ) ( y ) f 8 g 8 h 7 9 i 8-9 j ( ) ( 6m 8 ( ) k 6 ) - l m m - ( + ) n o + p q ( y ) y r ( p + ) ( p + ) + Indicil equtions We cn solve equtions of the form: Tke the cube of both sides: ( ) s follows: The left-hnd side becomes, so 8. However, when the unknown (or vrible) is not bse number but is n inde number, different pproch is required. Method : Ect solutions without clcultor To ttempt to solve inde equtions ectly, epress both sides of the eqution to the sme bse nd equte the powers. If m n, then m n (unless -, 0 or ).
13 Chpter 7 Eponentil nd logrithmic functions 69 8 Find the vlue of in ech of the following equtions. 8 b 6 c 6 6 Write the eqution. 8 Epress both sides to the sme bse. Equte the powers. Therefore,. b Write the eqution. b 6 Epress both sides to the sme bse. Equte the powers. Therefore,. Solve the liner eqution for by dding one to both sides. c Write the eqution. c Epress both sides to the sme bse. 6 (6 ) Remove the brckets by multiplying the powers. Equte the powers. Therefore, 6 Subtrct from both sides to mke 6 the subject. Add 6 to both sides to solve the eqution. More complicted equtions cn be solved using the sme technique. Solve for n in the following eqution: n 6 n Write the eqution. n 6 n + Epress both sides using the sme bse, n ( ) n +. Remove the brckets by multiplying the n n + powers. Multiply the terms on the left-hnd side 7n + by dding the powers. Equte the powers. Therefore, 7n + Solve the liner eqution for n. 7n n 7
14 70 Mths Quest Mths B Yer for Queenslnd In some cses indicil equtions cn be epressed in qudrtic form nd solved using the Null Fctor Lw. Look for numbers in inde form similr to nd ppering in different terms. Solve for if ( ) Write the eqution. () ( ) 0 Rewrite the eqution in qudrtic form. ( ) ( ) 0 Note tht ( ). Substitute y for. Let y. Rewrite the eqution in terms of y. Therefore, y y 0. Fctorise the left-hnd side. (y )(y + ) 0 Solve for y using the Null Fctor Lw. Therefore, y or y. Substitute for y. or Equte the powers. nd hs no solution. Stte the solution(s). Note tht in step 8, the possible solution ws rejected becuse there is no vlue of for which it will be stisfied. Recll tht eponentil functions such s re lwys positive. Method : Using clcultor nd tril nd error Indicil equtions which cnnot hve both sides epressed to the sme bse number do not generlly hve ect, rtionl solutions. A tril nd error method using clcultor cn find solutions to desired degree of ccurcy. Solve to deciml plces. Write the eqution. Get rough estimte of the solution. Since nd 8 then is between nd. Try. nd evlute too big, so try. Repet step until n estimte of desired ccurcy is found. Select the vlue of closest to...9 too smll, so try...78 too big, so try too big, so try...99 too smll, so try...08 too big, so try...0 too big Since.0 is too smll, nd. is too big,., to deciml plces.
15 Chpter 7 Eponentil nd logrithmic functions 7 Solving indicil equtions using the solve( commnd The solve( commnd is found in the CATALOG. The correct synt is: solve(epression, vrible, guess). The following steps show how to find the solution to the eqution.. Rerrnge the eqution so it is in the form f () 0. In this emple, we hve 0.. Press nd [CATALOG] (bove the zero key).. Press the S key (no need to first press ALPHA ).. Scroll down until you find solve(.. Press ENTER to pste solve( to the home screen. 6. Fill in the rguments of the solve function. In this emple, type the following: ^X,X,). Choose s our guess t the solution (not bd one, s, not long wy from 0!) If there re severl solutions, the solve function will return the one closest to the guess vlue. 7. Press ENTER. CASIO Solving indicil equtions remember remember. If m n, then m n (unless, 0 or ).. Inect solutions require the use of clcultor. 7C Indicil equtions 8 Solve for in ech of the following equtions. b 6 c d 0 e 6 f g - 8 h i 8 6 j k 8 l Eqution solver Mthcd Solve for n in ech of the following equtions. n + 6 b n + c n 7 8b d 6 n + e 9 n f 6 n 6 Find in ech of the following. 8 b c c d e + f g 9 + h Indicil equtions Mthcd
16 7 Mths Quest Mths B Yer for Queenslnd 9 0 Solve for in ech of the following equtions: 8 6 b c d e + + f 9 g h i j Solve for in ech of the following. ( ) + 0 b 6( ) c ( ) 6( ) d ( ) ( ) e ( ) ( ) f 0( ) + 0 g 6( ) 6 h ( ) + 0 ( ) Consider the indicil eqution ( ) The eqution cn be solved by mking the substitution: A y B y C y D y E y The qudrtic eqution formed by the pproprite substitution in question 6 is: A y y B y y C y + y D y y E y 9y The solutions to the eqution in question 7 re equls: A or B or C or D 0 or E 0 or 9 Solve ech of the following to deciml plces. b c 0 d 0 e 0 9 f 0 0 The nerest solution to the eqution 0 is: A. B. C.9 D E. Simulting rdioctivity Consider the imginry element Brggium. It is n unstble element nd every hour there is in 6 chnce tht n tom will decy. We will use rndom numbers to simulte the decy of Brggium. Suppose tht initilly there re 00 toms of Brggium. Generte rndom number from to 6 using die, clcultor or spredsheet. If the number is 6 this mens tht the tom decys. Repet this process 00 times.
17 Chpter 7 Eponentil nd logrithmic functions 7 After one hour, how mny toms hve decyed? After one hour, how mny toms of Brggium remin? Wht hppens to this smple of 00 toms by the end of the second hour? To simulte the decy of toms in the second hour, count the number of toms not yet decyed t the end of the first hour, n sy. Then generte n rndom numbers between nd 6. The number of 6s generted will give the number of toms tht hve decyed during the second hour. Copy the following tble nd complete it by generting rndom numbers between nd 6 s described bove. Time (t) in hours No. of toms remining (N) No. of toms decyed this hour Use the dt obtined in the tble for t 0 nd t to devise model for the simultion. Tht is, find vlues of nd b in N b t. Devise theoreticl model for the simultion. Tht is, find vlues of nd b in N b t given tht, in theory, in 6 of the Brggium toms decys ech hour. 6 Compre the theoreticl model devised in question with the dt. Which vlue of N in the tble differs most from the theoreticl prediction obtined in question?
18 7 Mths Quest Mths B Yer for Queenslnd Grphs of eponentil functions Functions of the form f (), where is positive rel number other thn nd is rel number, re clled eponentil functions. In generl, there re two bsic shpes for eponentil grphs: y, > or y, 0 < < y y y, > y, 0 < < 0 0 Incresing eponentil Decresing eponentil However, in both cses: the y-intercept is (0, ) the symptote is y 0 (-is) the domin is R the rnge is R +. Verify the shpes of these grphs by grphing, sy y, y, y nd y on grphics clcultor. The following sections on grphing eponentil functions show the rnge of vrition of eponentil grphs. Reflections of eponentil functions The grph of y is obtined by The grph of y is obtined by reflecting y through the y-is. reflecting y through the -is. y y, > y, > 0 y 0 y, > y, > Horizontl trnsltions of eponentil functions The grph of y + b is obtined by trnslting y :. b units to the right if b < 0. b units to the left if b > 0. For emple, the grph of y is obtined by trnslting y to the right units. Check this grph using grphics clcultor. Note lso tht ( )( ) ( ) so tht the effect is identicl to tht 8 of multiplying by constnt. y y y 0 units
19 Chpter 7 Eponentil nd logrithmic functions 7 Verticl trnsltions of eponentil functions The grph of y + c is obtined by trnslting y :. up by c units if c > 0. down by c units if c < 0. Furthermore the eqution of the symptote becomes y c. For emple, the grph of y 0 is obtined by trnslting y 0 down by units. The eqution of the symptote is y. Check this grph using grphics clcultor. units Find the eqution of the symptote nd the y-intercept. Hence, sketch the grph of y + nd stte its domin nd rnge. Write the rule. y + The grph is the sme s y trnslted units left nd units down. Stte the symptote. Asymptote is y. Evlute y when 0 to find the y-intercept. Locte the y-intercept nd symptote on set of es. Sketch the grph of the eponentil function using the y-intercept nd symptote s guide. y 6 0 y When 0, y Therefore, the y-intercept is (0, ). 0 y + y 0 y 0 (Asymptote) 7 Use the grph to stte the domin nd rnge. Domin is R. Rnge is (, ). Use grphics clcultor to solve using the intersection of two grphs. Give the nswer rounded to deciml plces. DISPLAY/ In the Y menu select Y nd enter ^X. Select Y nd enter. Set suitble WINDOW vlues. Press GRAPH. Press nd [CALC] nd select :intersect. 6 Press ENTER times (or follow the prompts). CASIO WE 7 7 Write the solution to deciml plces. Solution:.9
20 76 Mths Quest Mths B Yer for Queenslnd remember remember Generl shpes of grphs of eponentil functions: If f (), > If f (), 0 < < y y f(), > f(), 0 < < 0 0 In both cses, the y-intercept is (0, ) the symptote is y 0 the domin R the rnge R +. 7D Grphs of eponentil functions EXCEL SkillSHEET 7. Spredsheet Mthcd Eponentil functions Eponentil functions Sketch the grph of ech of the following on seprte es. (Use tble of vlues or copy grphics clcultor screen). y b y c y 6 d y 0 e y f y g y h y i y j y 0. k y.7 l y ( ) Sketch the following grphs, using tble of vlues or by copying grphics clcultor screen. Stte the eqution of the symptote nd the y-intercept for ech. y ( ) b y ( ) c y 0.( ) d y ( ) e y ( ) f y ( ) Find the eqution of the symptote nd the y-intercept for ech of the following. Hence, sketch the grph of ech nd stte its domin nd rnge. y b y + c y d y + e y f y + g y 6 + h y 0 + i y j y + + GC progrm Eponentil functions The rule for the grph t right is: A y B y C y D y + E y y 0
21 Chpter 7 Eponentil nd logrithmic functions 77 b The rule for the grph t right is: A y B y C y + D y + E y y 0 (, ) Use grphics clcultor to solve the following indicil equtions using the intersection of two grphs. Give nswers rounded to deciml plces. 0 b c 0.7 d 0 0 e 0 8 f 0 g 9 h i + j + WorkSHEET 7. A world popultion model EXCEL World popultion Spredsheet The sttistics below describe P, the estimted world popultion (in billions) t vrious times, t. t P t P
22 78 Mths Quest Mths B Yer for Queenslnd Use grphics clcultor or the Mths Quest Ecel file Eponentil model to plot the dt nd fit n eponentil curve. A not yet well fitted model is shown t right. If using grphics clcultor: Press STAT then select :Edit, nd enter the yers in L, nd the popultions in L. Press STAT, select [CALC] nd 0:EpReg, then nd L, nd L, Y nd ENTER. Y is found under VARS/Y-VARS/:Function. Use the eqution for the curve to predict the world popultion in 00. Wht limittions re there on the use of the eqution to predict future popultions? If using the spredsheet, comment on the effect of ech prt of the eqution on the shpe of the grph. Bode s Lw In 77, Johnn Bode discovered curious reltionship between pure numbers nd the distnce of plnets from the sun. His lw consisted of simple formul relting the number of the plnet to its distnce from the sun. The ctul distnces of the plnets from the sun re given in the tble below. By grphing the distnce ginst rised to the power of the plnet number, discover the reltionship tht Bode found. (Hint: Use either spredsheet or grphics clcultor to grph the dt nd then find the regression line.) Plnet number Plnet 0 Mercury 0.9 Venus 0.7 Erth Mrs. Ceres (steroid).77 Jupiter. 6 Sturn 9. 7 Urnus Pluto 9. Distnce in AU ( AU distnce from the Erth to the sun)
23 Chpter 7 Eponentil nd logrithmic functions 79 Logrithms The inde, power or eponent () in the indicil eqution y is lso known s logrithm. Logrithm y Bse numerl Bse This mens tht y cn be written in n lterntive form: log y which is red s the logrithm of y to the bse is equl to. For emple, 9 cn be written s log cn be written s log In generl, for > 0 nd : y is equivlent to log y. Using the indicil equivlent, it is possible to find the ect vlue of some logrithms. Evlute the following without clcultor. log 6 6 b log ( ) 8 Let equl the quntity we wish to find. Let log 6 6 Epress the logrithmic eqution s n indicil 6 6 eqution. Epress both sides of the eqution to the sme 6 6 bse. Equte the powers. b Write the logrithm s logrithmic eqution. b Let log ( ) Epress the logrithmic eqution s n indicil eqution. Epress both sides of the eqution to the sme bse. Equte the powers. 8 ( ) 8 Logrithm lws The inde lws cn be used to estblish corresponding rules for clcultions involving logrithms. These rules re summrised in the following tble. Nme Rule Restrictions Logrithm of product log (mn) log m + log n m, n > 0 > 0, Logrithm of quotient m, n > 0 log m - log m log n n > 0 nd Logrithm of power log m n n log m m > 0 > 0 nd Logrithm of the bse log > 0 nd Logrithm of one log 0 > 0 nd Logrithm lws Mthcd
24 80 Mths Quest Mths B Yer for Queenslnd It is importnt to remember tht ech rule works only if the bse,, is the sme for ech term. Note tht it is the logrithm of product nd logrithm of quotient rules tht formed the bsis for the pre-970s clcultion device for multipliction nd division the slide rule. Simplify, nd evlute where possible, ech of the following without clcultor. log 0 + log 0 b log + log 8 log Apply the logrithm of product rule. log 0 + log 0 log 0 ( ) Simplify. log 0 0 b Multiply the bse numerls of the logs being dded since their bses re the sme. b log + log 8 log log ( 8) log Apply the logrithm of quotient lw. log (96 ) Simplify, noting tht is power of. log log Evlute using the logrithm of power nd logrithm of the bse lws. log Simplify log log 0. Epress both terms s logrithms of log log 0 log log 0 inde numbers. Simplify ech logrithm. log log 00 Apply the logrithm of quotient lw. log ( 00) Simplify. log or log. Simplify ech of the following. log 8 9 log b log log 8 9 log Epress ech bse numerl s powers 8 7 log to the sme bse, 7. 8 log 8 7 log 8 7 Apply the logrithm of power lw. log 8 7 Simplify by cncelling out the common fctor of log 8 7. ( )
25 Chpter 7 Eponentil nd logrithmic functions 8 b Epress log 0 s log 0 nd s logrithm to bse 0 lso. Simplify using the logrithm of product lw. b log 0 + log 0 + log 0 0 log 0 0 remember remember. If y then log y where the bse, the power, inde or logrithm nd y the bse numerl. Note tht > 0,, nd therefore y > 0.. Log lws: () log m + log n log (mn) (b) log m log n log m - n (c) log m n n log m (d) log (e) log 0 7E Logrithms Epress the following indicil equtions in logrithmic form. 8 b c 0 d e b n f Epress the following logrithmic equtions in indicil form. log 6 b log c log d log 7 e log 6 f log 8 7 g log h log b 9-6 Logrithms to ny bse Mthcd The vlue of log is: A B C D E When epressed in logrithmic form, 8 is: A log 8 B log 8 C log 8 D log 8 E log 8 When epressed in indicil form, log is: A B C D E
26 8 Mths Quest Mths B Yer for Queenslnd 6 6 Evlute ech of the following without clcultor. log 6 b log 8 c log d log e log f log 0 ( ) g log 0. h log i log j log k log ( ) l log n n 7 Simplify, nd evlute where possible, ech of the following without clcultor. log 8 + log 0 b log 7 + log c log log 0 d log log 6 7 e log 0 log f log 6 log g log 00 log 8 h log + log 9 i log + log j log 0 log 0 0 k log log l log 9 + log log m log 8 log + log n log log log 6 8 Simplify ech of the following. - 6 log 0 + log 0 b log 8 + log c log + log d log log e log 0 + log 0 8 f log + log g log 7 log 6 h log ( ) + log i log 6 + log j log 0 ( + ) log 0 ( ) 9 Simplify the following. 7 log log 8 b - c log log 9 log 0 8 log d 7 e - f log 0 6 log 9 log 6 log g 0 h i log log 0 j log ( + ) log ( + ) log 6 - log 6 log - log log log 0 The epression log 0 y is equl to: A log 0 log 0 y B log 0 log 0 y C D y log 0 E log 0 + log 0 y log 0 log 0 y The epression log y is equl to: A log y B y log C log y D log + log y E y
27 Chpter 7 Eponentil nd logrithmic functions 8 The epression log 6 + log cn be simplified to: A log 0 B C log - D log 0 E log 6 0 The epression log log cn be simplified to: 7b A log B log C D log ( ) E log 7 Epress ech of the following in simplest form: log 7 + b log 6 + c log d + log 0 e log f log log 6 + g log 6 6 log 6 h + log 0 Solving logrithmic equtions Logrithms to the bse 0 Logrithms to the bse 0 re clled common logrithms nd cn be evluted using the LOG function on clcultor. For emple, to evlute log 0 8, correct to deciml plces:. Press 8 LOG on the clcultor nd press ENTER. (On some clcultors, press LOG 8.). The disply shows This mens log to deciml plces, or When solving logrithmic equtions involving bses other thn 0, the following steps should be followed: Step Simplify the eqution using logrithm lws. Step Step Epress the eqution in inde form if required. Solve by either: () evluting if the bse numerl is unknown (b) equting the powers if possible (c) equting the bses if possible. Note: The logrithm of negtive number or zero is not defined. Therefore: log is defined for > 0, if > 0 This cn be seen more clerly using inde nottion s follows: Let n log. Therefore, n (indicil equivlent of logrithmic epression). However, n > 0 for ll vlues of n if > 0 (positive bsed eponentils re lwys positive). Therefore, > 0.
28 8 Mths Quest Mths B Yer for Queenslnd Find if log 9. Write the eqution. log 9 Simplify the logrithm using the logrithm of power lw nd the fct tht log. log log Solve for by dding to both sides. Therefore, 8 Solve for if log 6. Write the eqution. log 6 Epress in inde form. Therefore, 6. Evlute the inde number Find if log, > 0. 0 Write the eqution. log Divide both sides by. log Write s n inde eqution. Therefore,. Epress both sides of the eqution to the sme bse,. Equte the bses. Note tht is rejected s solution, becuse > 0. Solving eponentil equtions using log 0 on the clcultor We hve lredy seen three methods for solving eponentil equtions:. equting the bses, which is not lwys n option, for emple, 7. using clcultor nd tril nd error, which cn be time consuming. using grphicl technique nd grphics clcultor. An efficient method for solving equtions involves the use of logrithms nd the log 0 function on the clcultor. This is outlined in the following emples.
29 Chpter 7 Eponentil nd logrithmic functions 8 Solve for, correct to deciml plces, if 7. 6 Write the eqution. 7 Tke log 0 of both sides. log 0 log 0 7 Use the logrithm of power lw to bring log 0 log 0 7 the power,, to the front of the logrithmic eqution. Divide both sides by log 0 to get by itself. log 0 7 Therefore, log Evlute the logrithms correct to deciml plces, t lest one more thn the nswer requires. Solve for..808 Therefore, we cn stte the following rule: If b, then log 0 b log 0 This rule pplies to ny bse, but since your clcultor hs bse 0, this is the most commonly used for this solution technique. remember remember. Logrithmic equtions re solved more esily by: () simplifying using log lws (b) epressing in inde form (c) solving s required.. If b, then log b 0. log 0 7F Solving logrithmic equtions 8 ORKED 9 Find in ech of the following. log b log 9 c log 7 d log 6 e log 0 - f log g log 8 h log i log j log + Solve for. log b log c log d log 0 e log 8 f log g log 6 h log 0 i log ( ) j log ( + ) k log 0 () l log 6 () m log log + log 6 n log log log log 8
30 86 Mths Quest Mths B Yer for Queenslnd 0 Solve for given tht: log 6 b log c log 6 6 d log e log 6 f log g log + 7 h log - The solution to the eqution log 7 is: A B C D 0 E b If log 8, then is equl to: A 096 B C 6 D E c Given tht log, must be equl to: A B 6 C 8 D E 9 d The solution to the eqution log log ( 8) is: A 8 B 6 C 9 D E Solve the following equtions correct to deciml plces. b 0.6 c 0 d.7 e 8 f 0.7 g 0 8 h + i + j + k 0 7 l The nerest solution to the eqution is: A 0.86 B. C. D E 0. WorkSHEET 7. 7 The nerest solution to the eqution 0.6 is: A 0.8 B 0. C 0.8 D 0.7 E 0. Logrithmic grphs Using grphics clcultor or grphing softwre, produce grphs of the following on the sme set of es. Ensure equl is scles if possible (if using grphics clcultor, use ZOOM nd :ZSqure). Copy the screen view into your workbook. y log 0 b y 0 c y Copy nd complete: The grph of y log 0 is the r of the grph of y 0 in the line y. Such functions re clled inverses of ech other. b An symptote is line tht grph never quite intersects. The line is n symptote for the grph of y log 0. Use technology to investigte the shpe of the grph of y Alog ( + b) + B for vrious vlues of the pronumerls A,, b nd B.
31 Chpter 7 Eponentil nd logrithmic functions 87 Sketch severl emples into your workbook, showing symptotes. The Mths Quest spredsheet Logrithmic grphs is idel for this. Wht is the effect of A on the grph? b Wht is the effect of on the grph? c Wht is the effect of b on the grph? d Wht is the effect of B on the grph? * Try sketching grphs of the following without using technology. (Hint: Find nd y intercepts by putting y 0 nd 0 respectively.) y log 0 ( ) + b y log 0 ( + ) c y log 0 EXCEL Logrithmic grphs Spredsheet Further work on logrithmic grphs is vilble on the Mths Quest CD-ROM. Click on the Etension Logrithmic grphs pnel. etension Logrithmic grphs The slide rule The logrithmic slide rule is compct device for rpidly performing clcultions with limited ccurcy. The invention of logrithms in 6 by John Npier mde it possible to multiply nd divide numbers by the more simple opertions of ddition nd subtrction. In this investigtion we will construct primitive slide rule.
32 88 Mths Quest Mths B Yer for Queenslnd Number Power of Tke two strips of crd bout cm by 0 cm. Mrk both crds s shown, using the numbers from the tble bove. You will notice tht the scle used is logrithmic scle using s bse. Tht is, the distnce from to 8 is units (log (8) ). Also, the distnce from to 0. is (log (0.) ). To multiply two numbers we need only to dd the powers so tht Thus, multiplying 8 by is equivlent to dding nd. The opertion of multipliction is converted to ddition. Your slide rule cn be used to perform this ddition. This slide rule is quite primitive nd in its 8. present form you would not use it to multiply by 0. However, this principle provided the bsis for scientific clcultions before the dvent of the electronic clcultor in the 960s nd 970s. Use your slide rule to clculte 0.. Use your slide rule to clculte. (Remember, division corresponds to subtrction of eponents.) Construct bse 0 slide rule Line up with the first fctor. 8 Red the result by reding the number corresponding to the second fctor.
33 Chpter 7 Eponentil nd logrithmic functions 89 Applictions of eponentil nd logrithmic functions Eponentil nd logrithmic functions cn be used to model mny prcticl situtions in science, medicine, engineering nd economics. A squre sheet of pper which is 0. mm thick is repetedly folded in hlf. Find rule which gives the thickness, T mm, s function of the number of folds, n. b Wht is the thickness fter 0 folds? c How mny folds re required for the thickness to rech 6 cm? T 0. when n 0 nd doubles with ech fold. This doubling implies tht the bse should be. Complete tble of vlues showing the thickness, T, for vlues of n from 0 to. Determine the rule for T(n). There is doubling term ( n ) nd multiplying constnt for the strting thickness (0.). Compre the rule for T(n) ginst the tble of vlues in step. When n 0, T 0. nd s n increses by, T doubles. n 0 T T(n) 0.( n ) b Substitute n 0 into the formul b When n 0, for T. T(0) 0.( 0 ) Clculte T. T 0. mm c Chnge 6 cm to millimetres. c 6 cm 60 mm Substitute T 60 into the formul. When T 60, ( n ) Divide both sides by n Tke log 0 of both sides. log log 0 n Use the logrithm of power lw log n log 0 to bring the power n to the front of the logrithm. 6 Divide both sides by log 0. log n log 0 7 Evlute. n 9. 8 Round the nswer up to the nerest whole number since the number of folds re positive integers nd if you round down the thickness will not hve reched 60 mm. Therefore, n 0 folds.
34 90 Mths Quest Mths B Yer for Queenslnd The price of gold P (dollrs per ounce) since 980 cn be modelled by the function: P log 0 (t + ), where t is the number of yers since 980. Find the price of gold per ounce in 980. b Find the price of gold in 999. c In wht yer will the price pss $0 per ounce? Stte the modelling function. P log 0 (t + ) Determine the vlue of t represented In 980, when t 0, by the yer 980. Substitute t into the modelling function. P log 0 [(0) + ] log 0 Evlute P. P 00 b Repet prt by determining the vlue of t represented by the yer 999. Substitute the vlue of t into the modelling function nd evlute P. b t t 9 When t 9, P log 0 [(9) + ] log $99. c Since P 0, substitute into the c log 0 (t + ) modelling function nd solve for t. Simplify by isolting the logrithm prt of the eqution log 0 (t + ). log 0 (t + ) Epress this eqution in its equivlent indicil form. 0 t + Solve this eqution for t..000 t t 99.8 t Convert the result into yers. The price will rech $0 in 99.8 yers fter 980. The price of gold will rech $0 in (pproimtely). remember remember. Red the question crefully.. Use the skills developed in the previous sections to nswer the question being sked.
35 Chpter 7 Eponentil nd logrithmic functions 9 7G Applictions of eponentil nd logrithmic functions Prior to mice plgue which lsts 6 months, the popultion of mice in country region is estimted to be The mice popultion doubles every month during the plgue. If P represents the mice popultion nd t is the number of months fter the plgue strts: b epress P s function of t find the popultion fter: i months ii 6 months c clculte how long it tkes the popultion to rech during the plgue. The popultion of town, N, is modelled by the function N 000( 0.0t ) where t is the number of yers since 980. Find the popultion in 980. b Find the popultion in: i 98 ii 990. c Wht is the predicted popultion in 00? d In wht yer will the popultion rech 0 000? The weight of bby, W kg, t weeks fter birth cn be modelled by W log 0 (8t + 0). Find the initil weight. b Find the weight fter: i week ii weeks iii 0 weeks. c Sketch the grph. d When will the bby rech weight of 7 kg? If $A is the mount n investment of $P grows to fter n yers t % p..: write A s function of P b use the function from to find the vlue of $0 000 fter 0 yers c clculte how mny yers it will be until n investment of $0 000 reches $6 00. The vlue of cr, $V, decreses ccording to the function V t. Find the vlue of the cr when new. b Find the vlue of the cr fter 6 yers. c In how mny yers will the cr be worth $0 000? 6 The temperture, T ( C), of cooling cup of coffee in room of temperture 0 C cn be modelled by T 90( 0.0t ), where t is the number of minutes fter it is poured. Find the initil temperture. b Find the temperture: i minutes fter pouring ii 6 minutes fter pouring. c How long is it until the temperture reches hlf its initil vlue? 7 A number of deer, N, re introduced to reserve nd its popultion cn be predicted by the model N 0(. t ), where t is the number of yers since introduction. Find the initil number of deer in the reserve. b Find the number of deer fter: i yers ii yers iii 6 yers. c How long does it tke the popultion to treble? d Sketch the grph of N versus t. e Eplin why the model is not relible for n indefinite time period.
36 9 Mths Quest Mths B Yer for Queenslnd 8 After recycling progrm is introduced the weight of rubbish disposed of by household ech week is given by W 80( 0.0t ), where W is the weight in kg nd t is the number of weeks since recycling ws introduced. Find the weight of rubbish disposed of before recycling strts. b Find the weight of rubbish disposed of fter recycling hs been introduced for: i 0 weeks ii 0 weeks. c How long is it fter recycling strts until the weight of rubbish disposed of is hlf its initil vlue? d i Will the model be relistic in 0 yers time? ii Eplin. 9 The number of hectres (N) of forest lnd destroyed by fire t hours fter it strted, is given by N 0 log 0 (00t + ). Find the mount of lnd destroyed fter: i hour ii hours iii 0 hours. b How long does the fire tke to burn out hectres? 0 A discus thrower competes t severl competitions during the yer. The best distnce, d metres, tht he chieves t ech consecutive competition is modelled by d 0 + log 0 (n), where n is the competition number. Find the distnce thrown t the: i st ii rd iii 6th iv 0th competition. b Sketch the grph of d versus n. c How mny competitions does it tke for the thrower to rech distnce of metres? The popultion, P, of certin fish t months fter being introduced to reservoir is P 00(0 0.08t ), 0 t 0. After 0 months, fishing is llowed nd the popultion is then modelled by P log 0 [0(t 9)], t 0. Find the initil popultion. b Find the popultion fter: i months ii months iii months iv 0 months. c How long does it tke the popultion to pss 0 000? A bll is dropped from height of metres nd rebounds to - of its previous height. 0 Find the rule tht describes the height of the bll (h metres) fter n bounces. b Find the height fter: i bounces ii 8 bounces. c Sketch the grph of the height of the bll fter n bounces. A computer pprecites in vlue by 0% per yer. If the computer costs $000 when new, find: the rule describing the vlue, V, of the computer t ny time, t yers, fter purchse. b the vlue of the computer fter 6 yers. c the number of yers it tkes to rech double its originl vlue. 7
37 Chpter 7 Eponentil nd logrithmic functions 9 At the beginning of this chpter we discussed the problem of deling, in meningful wy, with the lrge numbers tht rose in the mesurement of sound. We re now in position to propose solution to this problem. The solution involves the use of logrithmic scle. The decibel The loudness of sound, L, is mesured in decibels (db) nd is defined s follows: L 0 log 0 where I is the sound intensity mesured in wtts per squre metre (W/m ) nd I 0 is the threshold of hering nd hs vlue of 0 W/m. Intensity is not widely used to mesure sound becuse of the difficulty identified in the introduction to this chpter; tht is, the rnge in mgnitude from the threshold of humn hering ( 0 W/m ) to the sound of jckhmmer t distnce of 0 m (. 0 W/m ). The jckhmmer noise is. thousnd million times s intense s the softest sound. Such numbers re difficult to comprehend. The humn brin cn del with numbers tht rnge from 0 to 0, wheres it would struggle with. thousnd million. It is the logrithmic scle tht converts numbers tht hve lrge rnge to those tht re meningful to us. In prcticl situtions, we re usully interested in the effect of sound intensity on people. Clerly, sound level drops s we move wy from its source. Mesures of loudness my therefore need to show the distnce between the source of sound nd the observer. Consider the following dt. I - I 0 Sound Loudness in db Jet engine 0 Jckhmmer 90 Hevy trffic 7 Converstionl speech 60 Quiet living room 0 The threshold of pin for hering is db. How mny times s loud s jckhmmer is the pin threshold? Compre the intensity of the sound of converstion with tht of hevy trffic. How mny times is the sound of quiet living room s loud s tht of the threshold of humn hering?
38 9 Mths Quest Mths B Yer for Queenslnd Mthcd The Richter scle The Richter scle The Richter scle is used to describe the strength of erthqukes. A formul for the Richter scle is: R log K 0.9, where R is the Richter scle vlue for n erthquke tht releses K kilojoules (kj) of energy. Find the Richter scle vlue for n erthquke tht releses the following mounts of energy: 000 kj b 000 kj c 000 kj d kj e kj f kj. Does doubling the energy relesed double the Richter scle vlue? Find the energy relesed by n erthquke of: mgnitude on the Richter scle b mgnitude on the Richter scle c mgnitude 6 on the Richter scle. Wht is the effect (on the mount of energy relesed) of incresing the Richter scle vlue by? Why is n erthquke mesuring 8 on the Richter scle so much more devstting thn one tht mesures?
39 Chpter 7 Eponentil nd logrithmic functions 9 summry Inde lws m n m + n m n m n ( m ) n mn 0 (b) n n b n b n n b n Negtive nd rtionl powers n, 0 n n n m - n ( n ) m ( n n ) m m Indicil equtions If m n, then m n (unless, 0 or ). A grphics clcultor my be used to solve indicil equtions, using the solve( function. Grphs of eponentil functions If f (), > y 0 f(), > If f (), 0 < < y f(), 0 < < 0 y-intercept is (0, ) Asymptote is y 0 (-is) Domin R Rnge R +
40 96 Mths Quest Mths B Yer for Queenslnd Logrithms If y then log y where the bse, the power, inde or logrithm nd y the bse numerl. Log lws: log m + log n log (mn) m, n > 0 log m log n log m - m, n > 0 n log m n n log m m > 0 log log 0 Solving logrithmic equtions Logrithmic equtions re solved more esily by:. simplifying using log lws. epressing in inde form. solving s required. If log 0 b b, then. log 0 Logrithmic grphs The logrithmic function f () log is the inverse function of the eponentil function g(). y g(), > y f() log, > 0 -intercept is (, 0) Asymptote is y 0 Domin R + Rnge R
41 Chpter 7 Eponentil nd logrithmic functions 97 CHAPTER review ( y When simplified, ) y - is equl to: 7 y A y 7 B y 7 y - C 7 - D - E 7 7 y 6 y 7A m - p my be simplified to: m p ( m p6) m 7 p m A m - B 0 m - C m - D E 7 p 6 p 0 p 6 p 9 m 0 p 7A Simplify the following epression with positive indices. ( 6 6 y 0 ) ( 7 y 9 ) 7A,B The vlue of 0 6 is: A - B C D E 7B If, then is equl to: A B C D E 7C 6 If 7( ) + 6 0, then is equl to: A or 6 B 0 or C or 8 D or E 0 or 7C 7 Solve the following equtions. 00 b Find the solution to 9 ( ) correct to deciml plces. 7C 7C
42 98 Mths Quest Mths B Yer for Queenslnd 7D 9 The rule for the grph t right could be: A y B y + C y D y E y + y 0 7D Questions 0 to refer to the function defined by the rule y +. 0 The grph which best represents this function is: A y B y C 7 y D y 0 E 0 y D The domin is: A (, ) B [, ) C R + D R E R\{ } 7D 7D The rnge is: A [, ) B R C R + D (, ) E (, ) For the function with the rule f () + : find the y-intercept b stte the eqution of the symptote c sketch the grph of f () d stte the domin nd rnge. 7E When epressed in log form, 0 becomes: A log 0 B log 0 C log 0 D log 0 E log 0 7E 7E The vlue of log log 8 is: A B 7 C 0 D 69 E 6 Evlute log -. 7 b Epress y in terms of if log 0 + log 0 y log 0 ( + ).
43 Chpter 7 Eponentil nd logrithmic functions 99 7 Simplify the following. log log 6 log b - log 7E 8 log The vlue of - is nerest to: log A B C D 9 E 0 7F 9 The epression log 7 log 7 simplifies to: 7F A B log 7 C log 7 D E 8 0 The solution to log is: A B C D 6 E 0 7F The vlue of if log 6 is: 7F A B 7 C D E 7 If log ( ) + log, then is equl to: A B C D E - 7F The solution to the eqution is nerest to: A B 0. C 0.60 D 0. E 0 7F Solve ech of the following. log 6 b log 6 c log ( + 6) log If, find correct to deciml plces. 6 Solve for where F 7F 7F
44 00 Mths Quest Mths B Yer for Queenslnd 7G 7 The number of bcteri in culture, N, is given by the eponentil function N 00( 0.8t ), where t is the number of dys. Find the initil number of bcteri in the culture. b Find the number of bcteri (to the nerest 00) fter: i dys ii 0 dys. c How mny dys does it tke for the number of bcteri to rech 9000? 8 The number of lions, L, in wildlife prk is given by L 0(0 0.t ), where t is the number of yers since counting strted. At the sme time the number of cheeths, C, is given by C (0 0.0t ). Find the number of: i lions ii cheeths when counting begn. b Find the numbers of ech fter i yer ii 8 months. c Which of the nimls is the first to rech popultion of 0 nd by how long? d After how mny months re the popultions equl nd wht is this popultion? test yourself CHAPTER 7
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