Eponentil nd logrithmic functions VCE co covverge Ares of study Unit Functions nd grphs Algebr In this ch chpter pter A Inde lws B Negtive nd rtionl powers C Indicil equtions D Grphs of eponentil functions E Logrithms F Solving logrithmic equtions G Applictions of eponentil nd logrithmic functions
68 Mthemticl Methods Units nd Introduction Functions in which the independent vrible is n inde number re clled indicil or eponentil functions. For emple: f () where > 0 nd is n eponentil function. It cn be shown tht quntities which increse or decrese by constnt percentge in prticulr time cn be modelled by n eponentil function. Eponentil functions hve pplictions in science nd medicine (eponentil decy of rdioctive mteril, eponentil growth of bcteri like those shown in the photo), nd finnce (compound interest nd reducing blnce lons). Inde lws We recll tht number,, which is multiplied by itself n times cn be represented in inde nottion. Inde (or power or eponent)... n Bse 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. For emple, 7 Division When dividing two numbers in inde form with the sme bse, subtrct the indices. For emple, 6 Rising to power To rise n indicil epression to power, multiply the indices. For emple, ( ) + + m n m + n m n m n ( m ) n m n mn 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.
Chpter Eponentil nd logrithmic functions 69 Products nd quotients Note the following. For emple, ( ) ( ) ( ) ( ) ( ) Simplify. y y b ( y ) y c () b 6 9 b d Collect plin numbers ( nd ) nd terms with the y y y y sme bse. Simplify by multiplying plin 8 y numbers nd dding powers with the sme bse. b Remove the brcket by multiplying the powers. (The b ( y ) y y 6 y power of the inside the brcket is.) Convert to plin number () first nd collect terms with y 6 y the sme bse. Simplify by dding powers with the sme bse. y 0 c Write the quotient s frction. c () b 6 9 b ( ) b 6 9 b d Remove the brcket by multiplying the powers. b - 6 9 b (b) n n b n b n 8 p 6 m ( p) m 6 p m 7 Simplify by first cncelling b 6 plin numbers. b Complete simplifiction by 7b subtrcting powers with the sme bse. (Note:.) 8 p Epnd the brckets by rising d 6 m ( p) m 8 p - 6 m p m ech term to the power of. 6 p m 6 p m Convert 8 7 p to 7 nd collect 6 p m m - like pronumerls. 6 p m Simplify by first reducing the 6p 6 + m + plin numbers, nd the pronumerls by dding the indices for multipliction nd subtrcting the indices for division. Simplify the indices of ech bse. 6p m 6 - n b n
70 Mthemticl Methods Units nd Simplify 6 6 b - 6 7 b 6 b b Write the epression Chnge the division sign to multipliction nd replce the second term with its reciprocl (turn the second term upside down). 6 b 6 7 b 6 6 Remove the brckets by multiplying the b 9 b 6 powers. 6 7 b 6 6 b 6 8 Collect plin numbers nd terms with the + 9 7 6 b + 6 6 6 7 sme bse. Cncel plin numbers nd pply inde lws. 0 b 0 9 Simplify. b - b 6 b 6 7 b 6 9 b - b Epressions involving just numbers nd numericl indices cn be simplified using inde lws nd then evluted. Write in simplest inde nottion nd evlute. 6 b 9 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 Rewrite the bses in terms of their prime fctors. Simplify the brckets using inde nottion. Remove the brckets by multiplying the 9 b ( ) 7 ( ) ( ) - ( ) 0 powers. Write in simplest inde form. Evlute s bsic number. 9
Chpter Eponentil nd logrithmic functions 7 Comple epressions involving terms with different bses hve to be simplified by replcing ech bse with its prime fctors. Simplify n 8 - n +. 6 n Rewrite the bses in terms of their prime fctors. Simplify the brckets using inde nottion. Remove the brckets by multiplying powers. Collect terms with the sme bse by dding the powers in the products nd subtrcting the powers in the quotients. Simplify. n 8 n + n ( ) n + 6 n ( ) n n ( ) n + ( ) n n n + n + n n n + n + ( n ) n + ( n ) 6n + n + n + n + n + n + n + n 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)
7 Mthemticl Methods Units nd A Inde lws Mthcd Indices, b c Simplify. 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. 7 b 8 b b b 9 () b c ( )y 6 y d p q 0 (pq ) e (mn ) n f b b g r s 0 t r (s ) (t) h 6 b 7 b i y 0 y Simplify. d 6 p 8 m p 7 m 6 9 p m u c v 9 ( u ) v u 6 v 6w e t 7 9w t ( w) t ( g y ) - y 6 7 y 6 ( y) m i p ( mp ) ( mp ) Simplify. 8 b b 9 b b - b k d - ( k ) 6kd ( k d ) c e g ( p ) - g p 8g p 7 ( gp) y 7 y y y b d f h j d f ( ) y 6 y 0 7 y ( e ) f 8e f 0ef ( ) y ( 7 y) - 8 ( y) ( mp) m p - ( mp) ( u 7 v 6 ) - ( u v ) u ( v ) jn ( j n) n n ( j) 6 y 8 ( y - ) ( y ) 8 y 7
Chpter Eponentil nd logrithmic functions 7 multiple choice p m - cn be simplified to: p m A p m B p m 6 C p m 8 D p m E 6 b 6 y - cn be simplified to: y ( y) A y B y C 0 y 6 D 9 y 0 - E - b c - is equl to: b b A 8 B 6 C 6 b D - E 6 6 Simplify ech of the following. n + y z n ( y n z n b nym + ) y - 7 Write in simplest inde nottion nd evlute. 8 b 7 9 7 8 c d 0 8 e 7 - f 6 8 Write in simplest inde nottion nd evlute. b c d 7 9 ( 6 7 ) ( ) ( 6) e ( ) - f - g 8 h ( ) ( ) 6 9 Simplify. n 9 - n + c e n 6 n - 6 n 7 n 9 n + n n n + g 6 n 9 - h * + 8 n 6 n i * n - n + n + + n *Hint: Fctorise the numertor nd denomintor first. b d f n + y m b 6 n n n + n 7 n 9 n 8 n 6 n 7-6 9 n + - n + n + n n y m 8 0 7 - ( ) 7 8-9 0 multiple choice 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
7 Mthemticl Methods Units nd 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 -, 0 0 n using division rule for indices n simplifying the inde. In other words, n n - nd - n n Note: Chnge the level, chnge the sign. Epress ech of the following with positive inde numbers. b 8 y ( 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. 8 8 8 8 b y ( y) y 0 y y 6 y 7 y 6 y ( ) y 7 y 0 y 0
Chpter Eponentil nd logrithmic functions 7 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, 6 m n - + ( n) m ( Evlute ech of the following without clcultor. 6 b 9 - n n ) m m Rewrite the bse number in terms of its 6 ( ) prime fctors. Remove the brckets by multiplying the 6 powers. Evlute s bsic number. 6 b Rewrite the bse numbers of the 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. 9 b - 7
76 Mthemticl Methods Units nd 7 Simplify the following epressing your nswer with positive indices. 7 6 b y 6 y Write the epression. 7 6 Write using frctionl indices 6 7 Write 6 nd in inde form. ( 7 ) ( ) Multiply the powers. 6 Simplify the powers. 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 Collect terms with the sme bse by subtrcting the powers. y Simplify the powers. 6 y 6 Rewrite with positive powers. - 6 y 7 6 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
Chpter Eponentil nd logrithmic functions 77 B Negtive nd rtionl powers Epress ech of the following with positive inde numbers. 6 b c ( ) 7 d ( ) e ( ) 9 f ( 6 ) g ( ) h SkillSHEET. b Simplify ech of the following, epressing your nswer with positive inde numbers. ( ) ( b ) ( y ) - c ( y ) d ( ) ( e ) ( ) - f ( ) ( ) ( ) ( m) m - ( p ) p y ( y ) ( ) ( y ) Negtive nd rtionl powers Mthcd Evlute the following without clcultor. 6 9 b 7 c 6 d e 8 f 8 g 8 h 6 8 6 7 6 i - j - k - l m n - o p multiple choice The ect vlue of 6 is: A B 6 C D - E b The ect vlue of - is: 8 A B 6 C D E c simplifies to: 6 8 8 8 6 7 6 7 6 6-6 6 7 6 A B C D E - 6-6 Simplify ech of the following, epressing your nswer with positive indices. 7 9 8 b 6 c d ( ) e ( y ) ( y ) f g 8 h 7 9 i j ( ) ( 6m 8 ( ) k 6 ) - l m ( + ) m - n o + p + + - q ( y ) y r ( p + ) ( p + ) + 9 8 8 8-9
78 Mthemticl Methods Units nd Indicil equtions We cn solve equtions of the form: s follows: Tke the cube of both sides: ( ) 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. 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 6 6 6 Epress both sides to the sme bse. 6 (6 ) Remove the brckets by multiplying the powers. 6 6 6 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 + 6 Chpter Eponentil nd logrithmic functions 79 9 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 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 ( ) 0. 6 7 8 9 0 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.
80 Mthemticl Methods Units nd Solve to deciml plces. Write the eqution. Get rough estimte of the solution. Since nd 8 then is between nd. Try. nd evlute....67 too big, so try. Repet step until n estimte of desired ccurcy is found...9 too smll, so try...78 too big, so try...098 too big, so try...99 too smll, so try...08 too big, so try...0 too big Select the vlue of closest to. Since.0 is too smll, nd. is too big,., to deciml plces. 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,) is 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. remember remember. If m n, then m n.. Inect solutions require the use of clcultor.
Chpter Eponentil nd logrithmic functions 8 C Indicil equtions 8 8b 9 0 Solve for in ech of the following equtions. b 6 c d 0 e 6 f 6 Solve for n in ech of the following equtions. n + 6 b n + c n 7 d 6 n + e 9 n f 6 n 6 Find in ech of the following. 8 b 7 9 + c 6 + 8 8c d e + f 6 6 + Solve for in ech of the following equtions: 8 6 b c 7 + 8 d 6 + + g 9 7 + h 7 + 6 + i 6 + 6 - j 6 6 6 Solve for in ech of the following. ( ) + 0 b 6( ) + 8 0 c ( ) 6( ) + 96 0 d ( ) ( ) + 0 0 e ( ) ( ) f 0( ) + 0 g 6( ) 6 h ( ) + 0 ( ) 6 7 8 multiple choice Consider the indicil eqution ( ) + 7 0. The eqution cn be solved by mking the substitution: A y B y C y D y E y multiple choice The qudrtic eqution formed by the pproprite substitution in question 6 is: A y y + 7 0 B y y + 7 0 C y + y + 7 0 D y y + 7 0 E y 9y + 0 multiple choice 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 00 8 g - h i 8 6 j k 8 l g 9 + h 6 e + + f 9 multiple choice The nerest solution to the eqution 0 is: A. B. C.9 D E. 7-8 + 8 7 + 6 Eqution solver Indicil equtions Mthcd Mthcd
8 Mthemticl Methods Units nd 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 Incresing eponentil 0 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 go beyond the requirements of the study design, but re included to show the rnge of vrition of eponentil grphs. Reflections of eponentil functions The grph of y is obtined by reflecting y through the y-is. y y, > 0 y, > The grph of y is obtined by reflecting y through the -is. 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. 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. 0 y, > y, > units y y y 0 0 y units y 0 y 0 (Asymptote)
Chpter Eponentil nd logrithmic functions 8 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 When 0, y Therefore, the y-intercept is (0, ). 0 y + 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). 7 Write the solution to deciml plces. Solution:.9 remember remember Generl shpes of grphs of eponentil functions: If f (), > If f (), 0 < < y y f(), > 0 f(), 0 < < 0 In both cses, the y-intercept is (0, ) the symptote is y 0 the domin R the rnge R +.
8 Mthemticl Methods Units nd D Grphs of eponentil functions EXCEL SkillSHEET. 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 multiple choice The rule for the grph t right is: A y B y C y D y + E y y 0 b The rule for the grph t right is: A y B y C y + D y + E y y 0 (, ) WorkSHEET. 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 +
Chpter Eponentil nd logrithmic functions 8 A world popultion model EXCEL World popultion Spredsheet The sttistics below describe P, the estimted world popultion (in billions) t vrious times t. t 0 000 0 00 70 800 80 900 90 P 0.0 0. 0.0 0.0 0.79 0.98.6.6.7 t 90 90 90 90 960 970 980 990 000 P.86.07.0..0.70..0 6. 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.
86 Mthemticl Methods Units nd Logrithms Logrithm The inde, power or eponent () in the indicil y eqution y is lso known s logrithm. 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 9. 0 00 000 cn be written s log 0 00 000. In generl, for > 0 nd : y is equivlent to log y. Creer profile ALISON HENNESSY Audiologist Qulifictions: Bchelor of Science Grdute Diplom in Audiology Mster of Science Employer: Bionic Er Institute, Est Melbourne Compny website: http://www.medoto.unimelb.edu.u/oto/ http://www.medoto.unimelb.edu.u/crc/ Audiology ppeled to me s it combined my love of science nd my desire to work with people. Prt of my dy involves clinicl work: people ttending our clinic for hering tests, specilised dignostic tests, counselling bout hering loss, nd for the fitting nd evlution of hering ids. I lso give lectures to our university students nd m involved in ttrcting reserch students to our Coopertive Reserch Centre. I red to keep up-to-dte with dvnces in hering ids. I use number of different formuls for prescribing the mplifiction required from hering ids for hering-impired clients. The mount of mplifiction is dependent on number of fctors: the hering loss t prticulr frequency, the hering loss t other frequencies, the growth of loudness nd the listening sitution (for emple, quiet or noisy, soft or loud sound). Formuls re becoming more complicted s hering id technology improves. Another re where formul is used is in the clcultion of sound intensity levels, which re mesured in decibels. One formul is: L 0 log I 0 - where L is the sound intensity I0 level in decibels (db), I is the sound intensity in wtts per squre metre (W/m ) nd I 0 is the reference sound intensity which is usully the threshold of hering nd so hs vlue of.0 0 W/m. To interpret journl rticles nd mrketing mterils sent by hering id mnufcturers, I need to understnd the mthemticl processes involved in the sttisticl tests reported. This knowledge helps me to ssess whether the clims re supported by the results. Mths is involved in subtle wy in my job. The emphsis is on correctly ssessing hering loss nd providing the best possible ssistnce in improving communiction. In order to do this I spend lot of time tlking to people nd conducting tests. At the sme time it is criticl to the success of my job tht I understnd scientific principles nd mthemticl processes. Questions. List two resons why Alison finds mthemtics useful in her profession.. Use the formul given to clculte the sound intensity level for jckhmmer, t distnce of 0 m, tht hs n intensity of. 0 W/m.. Find out wht subjects re recommended to be undertken in Yer to continue on to tertiry course in this field.
Chpter Eponentil nd logrithmic functions 87 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 ( ) Let equl the quntity we wish to find. Let log 6 6 Epress the logrithmic eqution s n 6 6 indicil eqution. Epress both sides of the eqution to 6 6 the sme bse. Equte the powers. b 8 Write the logrithm s logrithmic eqution. Epress the logrithmic eqution s n indicil eqution. Epress both sides of the eqution to the sme bse. Equte the powers. b Let log ( ) 8 ( ) 8 The following section goes beyond the requirements of the study design but is importnt prticulrly for Mthemticl Methods Units &. 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 lws Mthcd 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 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.
88 Mthemticl Methods Units nd 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 log 0 + log 0 log 0 ( ) rule. 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 log 96 log Apply the logrithm of quotient log (96 ) lw. 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 b log 0 + log 8 b 6 Epress ech bse numerl s powers to the sme bse, 7. log 8 9 log 8 7 log 8 log 8 7 Apply the logrithm of power lw. log 8 7 log 8 7 Simplify by cncelling out the common fctor of log 8 7. Epress log 0 s log 0 nd s logrithm to bse 0 lso. b log 0 + log 0 + log 0 0 Simplify using the logrithm of product lw. log 0 0 7 ( )
remember remember Chpter Eponentil nd logrithmic functions 89. 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 E Logrithms Epress the following indicil equtions in logrithmic form. 8 b c 0 d 0.0 0 e b n f Epress the following logrithmic equtions in indicil form. log 6 b log 0 000 000 6 c log d log 7 e log 6 f log 8 7 g log h log b 9 multiple choice The vlue of log is: A B C D E multiple choice When epressed in logrithmic form, 8 is: A log 8 B log 8 C log 8 D log 8 E log 8 multiple choice When epressed in indicil form, log 0 0 000 is: A 0 0 000 B 0 000 0 C 0 000 0 D 0 0 000 E 0 0 000 6 Evlute ech of the following without clcultor. log 6 b log 8 c log d log e log 0 000 f log 0 (0.000 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 0 0 + log 0 d log 6 8 + 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 6-6 - 6 Logrithms to ny bse Mthcd
90 Mthemticl Methods Units nd 6 8 Simplify ech of the following. 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 multiple choice 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 multiple choice The epression log y is equl to: A log y B y log C log y D log + log y E y multiple choice The epression log 6 + log cn be simplified to: A log 0 B C log - D log 0 E log 6 0 multiple choice log The epression 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
Chpter Eponentil nd logrithmic functions 9 Solving logrithmic equtions This section goes beyond the study design requirements but could be considered s preprtion for Mthemticl Methods Unit. 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 0.90 089 887. This mens log 0 8 0.90 to deciml plces, or 0 0.90 8. When solving logrithmic equtions involving bses other thn 0 the following steps should be followed:. Simplify the eqution using log lws.. 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. 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. - 9 6-6
9 Mthemticl Methods Units nd 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. 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 0 0.8 Evlute the logrithms correct to deciml - 0.00 plces, t lest one more thn the nswer requires. Solve for..808 Therefore, we cn stte the following rule: If log 0 b b, then. 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 Chpter Eponentil nd logrithmic functions 9. 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 F Solving logrithmic equtions 8 9 0 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 8 0 9 i log 0 000 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 Solve for given tht: log 6 b log c log 6 6 multiple choice 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 8 0.7 6 7 d log e log 6 f log 6 0 00 g log + 7 h log - multiple choice The nerest solution to the eqution is: A 0.86 B. C. D E 0. multiple choice The nerest solution to the eqution 0.6 is: A 0.8 B 0. C 0.8 D 0.7 E 0. WorkSHEET.
9 Mthemticl Methods Units nd Logrithmic grphs EXCEL Spredsheet 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 work book. 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. The following is beyond the scope of the study design for units nd Mthemticl Methods, but is still of interest here. Use technology to investigte the shpe of the grph of y Alog ( + b) + B for vrious vlues of the pronumerls A,, b nd B. 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.) log 0 ( ) + b log 0 ( + ) c y log 0 Further work on logrithmic grphs is vilble on the Mths Quest CD Rom. Click on the Etension Logrithmic grphs pnel. etension Logrithmic grphs
Chpter Eponentil nd logrithmic functions 9 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 0. 0. 0. 0.8.6. 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, 60 0. ( n ) Divide both sides by 0.. 600 n Tke log 0 of both sides. log 0 600 log 0 n Use the logrithm of power lw log 0 600 n log 0 to bring the power n to the front of the logrithm. 6 Divide both sides by log 0. log 0 600 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.
96 Mthemticl Methods Units nd The price of gold P (dollrs per ounce) since 980 cn be modelled by the function: P 00 + 0 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 00 + 0 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 00 + 0 log 0 [(0) + ] 00 + 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 999 980 t 9 When t 9, P 00 + 0 log 0 [(9) + ] 00 + 0 log 0 96 00 + 99. $99. c Since P 0, substitute into the c.0 00 + 0 log 0 (t + ) modelling function nd solve for t. Simplify by isolting the logrithm prt of the eqution.. 0 0 log 0 (t + ). log 0 (t + ) Epress this eqution in its equivlent indicil form. 0 t + Solve this eqution for t..000 t +.999 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 980 + 99.8 80 (pproimtely). remember remember. Red the question crefully.. Use the skills developed in the previous sections to nswer the question being sked.
Chpter Eponentil nd logrithmic functions 97 G 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 0 000. 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 c epress P s function of t find the popultion fter: i months ii 6 months clculte how long it tkes the popultion to rech 00 000 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 000 0.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.
98 Mthemticl Methods Units nd 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 000 + 9 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? 7 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.
Chpter Eponentil nd logrithmic functions 99 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. The Richter scle Mthcd Find the Richter scle vlue for n erthquke tht releses the following mounts of energy: 000 kj b 000 kj c 000 kj d 0 000 kj e 00 000 kj f 000 000 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?
00 Mthemticl Methods Units nd 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. 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 +
Chpter Eponentil nd logrithmic functions 0 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
0 Mthemticl Methods Units nd CHAPTER review A A Multiple choice ( y When simplified, ) y - is equl to: 7 y A y 7 B y 7 y - C 7 - D - E 7 7 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 y 6 y m 0 p B 6 The vlue of is: 0 A - B C D E C If, then is equl to: A B C D E C D If 7( ) + 6 0, then is equl to: A or 6 B 0 or C or 8 D or E 0 or 6 The rule for the grph t right could be: A y B y + C y D y E y + y 0 D Questions 7 to 9 refer to the function defined by the rule y +. 7 The grph which best represents this function is: A y B y C 7 y D y 0 E 0 y 0 0 0
Chpter Eponentil nd logrithmic functions 0 8 The domin is: A (, ) B [, ) C R + D R E R\{ } 9 The rnge is: A [, ) B R C R + D (, ) E (, ) 0 When epressed in log form, 0 becomes: A log 0 B log 0 C log 0 D log 0 E log 0 The vlue of log 7 9 + log 8 is: A B 7 C 0 D 69 E log The vlue of - is nerest to: log A B C D 9 E 0 log 7 The epression log 7 simplifies to: D D E E F F A B log 7 C log 7 D E 8 The solution to log is: A B C D 6 E 0 F The vlue of if log 6 is: A B 7 C D E 6 If log ( ) + log, then is equl to: A B C D E - 7 F F 7 The solution to the eqution is nerest to: A B 0. C 0.60 D 0. E 0 F Short nswer Simplify the following epression with positive indices. ( 6 6 y 0 ) ( 7 y 9 ) Solve the following equtions. 00 b 8 + Find the solution to 9 ( ) + 6 0 correct to deciml plces. A,B C C
0 Mthemticl Methods Units nd D E E F F F G 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. Evlute log - 7 b Epress y in terms of if log 0 + log 0 y log 0 ( + ). 6 Simplify the following. log log 6 b log - log 7 Solve ech of the following. log 6 b log 6 c log ( + 6) log 8 If, find correct to deciml plces. 9 Solve for where 0 8. 0 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? Anlysis 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