Section 4.3 Logarithmic Functions

Transcription

1 48 Chapr 4 Sion 4.3 Logarihmi Funions populaion of 50 flis is pd o doul vry wk, lading o a funion of h form f ( ) 50(), whr rprsns h numr of wks ha hav passd. Whn will his populaion rah 500? Trying o solv his prolm lads o: () Dividing oh sids y 50 o isola h ponnial 0 Whil w hav s up ponnial modls and usd hm o mak prdiions, you may hav noid ha solving ponnial quaions has no y n mniond. Th rason is simpl: non of h algrai ools disussd so far ar suffiin o solv ponnial quaions. Considr h quaion 0 aov. W know ha 3 8 and 4 6, so i is lar ha mus som valu wn 3 and 4 sin g ( ) is inrasing. W ould us hnoy o ra a al of valus or graph o r sima h soluion. From h graph, w ould r sima h soluion o around 3.3. This rsul is sill fairly unsaisfaory, and sin h ponnial funion is on-o-on, i would gra o hav an invrs funion. Non of h funions w hav alrady disussd would srv as an invrs funion and so w mus inrodu a nw funion, namd as h invrs of an ponnial funion. Sin ponnial funions hav diffrn ass, w will dfin orrsponding arihms of diffrn ass as wll. Logarihm Th arihm (as ) funion, wrin funion (as ),., is h invrs of h ponnial Sin h arihm and ponnial ar invrss, i follows ha: Propris of Logs: Invrs Propris

2 Sion 4.3 Logarihmi Funions 49 Rall also from h dfiniion of an invrs funion ha if pplying his o h ponnial and arihmi funions: f ( a), hn f ( ) a. Logarihm Equivaln o an Eponnial Th samn a is quivaln o h samn ( ) a. a lrnaivly, w ould show his y saring wih h ponnial funion, hn a aking h as of oh sids, giving ( ). Using h invrs propry of s w s ha ( ) a. Sin is a funion, i is mos orrly wrin as ( ), using parnhss o dno funion valuaion, jus as w would wih f(). Howvr, whn h inpu is a singl varial or numr, i is ommon o s h parnhss droppd and h prssion wrin as. Eampl (vido ampl hr) Wri hs ponnial quaions as arihmi quaions: is quivaln o (8) is quivaln o 5 (5) 0 is quivaln o Eampl (vido ampl hr) Wri hs arihmi quaions as ponnial quaions: is quivaln o is quivaln o Try i Now Wri h ponnial quaion 4 6 as a arihmi quaion.

3 50 Chapr 4 By salishing h rlaionship wn ponnial and arihmi funions, w an now solv asi arihmi and ponnial quaions y rwriing. Eampl 3 4 Solv (vido ampl hr) for. By rwriing his prssion as an ponnial, 4, so = 6 Eampl 4 Solv 0 for. By rwriing his prssion as a arihm, w g (0) Whil his dos dfin a soluion, and an a soluion a ha, you may find i somwha unsaisfying sin i is diffiul o ompar his prssion o h dimal sima w mad arlir. lso, giving an a prssion for a soluion is no always usful ofn w rally nd a dimal approimaion o h soluion. Lukily, his is a ask alulaors and ompurs ar qui adp a. Unlukily for us, mos alulaors and ompurs will only valua arihms of wo ass. Happily, his nds up no ing a prolm, as w ll s rifly. Common and Naural Logarihms Th ommon is h arihm wih as 0, and is ypially wrin ( ). Th naural is h arihm wih as, and is ypially wrin ln( ). Eampl 5 Evalua ( 000) using h dfiniion of h ommon. To valua ( 000), w an say (000), hn rwri ino ponnial form using h ommon as of From his, w migh rogniz ha 000 is h u of 0, so = 3. W also an us h invrs propry of s o 3 0 wri 3 0 Valus of h ommon numr numr as (numr) ponnial

4 Sion 4.3 Logarihmi Funions 5 Try i Now. Evalua ( ). Eampl 6 Evalua ln. W an rwri ln as ln / propry for s: ln / /. Sin ln is a as, w an us h invrs. Eampl 7 Evalua (500) using your alulaor or ompur. Using a ompur, w an valua ( 500) To uiliz h ommon or naural arihm funions o valua prssions lik (0), w nd o salish som addiional propris. Propris of Logs: Eponn Propry r r To show why his is ru, w offr a proof. Sin h arihmi and ponnial funions ar invrss, r So r Uilizing h ponnial rul ha sas r r r r r So hn p q pq, gain uilizing h invrs propry on h righ sid yilds h rsul r r. Eampl 8 (vido ampl hr) Rwri 3 5 using h ponn propry for s. No, h vido in his ampl is an ovrviw of many arihmi propris. To viw jus h propry for his ampl, skip o 3 minus and 6 sonds ino h vido. Sin 5 = 5,

5 5 Chapr 4 Eampl 9 Rwri 4ln( ) using h ponn propry for s. 4 Using h propry in rvrs, 4ln( ) ln Try i Now 3. Rwri using h ponn propry for s: ln. Th ponn propry allows us o find a mhod for hanging h as of a arihmi prssion. Propris of Logs: Chang of Bas ( ) ( ) Proof: L. Rwriing as an ponnial givs. Taking h as of oh sids of his quaion givs Now uilizing h ponn propry for s on h lf sid, Dividing, w oain or rplaing our prssion for, Wih his hang of as formula, w an finally find a good dimal approimaion o our qusion from h ginning of h sion. Eampl 0 (vido ampl hr) Evalua (0) using h hang of as formula. ording o h hang of as formula, w an rwri h as as a arihm of any ohr as. Sin our alulaors an valua h naural, w migh hoos o us h naural arihm, whih is h as : 0 ln0 0 ln Using our alulaors o valua his, ln ln This finally allows us o answr our original qusion h populaion of flis w disussd a h ginning of h sion will ak 3.3 wks o grow o 500.

6 Sion 4.3 Logarihmi Funions 53 Eampl (vido ampl hr) Evalua 5 (00) using h hang of as formula. W an rwri his prssion using any ohr as. If our alulaors ar al o valua h ommon arihm, w ould rwri using h ommon, as (00) Whil w wr al o solv h asi ponnial quaion 0 y rwriing in arihmi form and hn using h hang of as formula o valua h arihm, h proof of h hang of as formula illuminas an alrnaiv approah o solving ponnial quaions. Solving ponnial quaions:. Isola h ponnial prssions whn possil. Tak h arihm of oh sids 3. Uiliz h ponn propry for arihms o pull h varial ou of h ponn 4. Us algra o solv for h varial. Eampl Solv 0 (vido ampl hr) for. Using his alrnaiv approah, rahr han rwri his ponnial ino arihmi form, w will ak h arihm of oh sids of h quaion. Sin w ofn wish o valua h rsul o a dimal answr, w will usually uiliz ihr h ommon or naural. For his ampl, w ll us h naural : ln ln(0) Uilizing h ponn propry for s, ln ln(0) Now dividing y ln(), ln(0).86 ln Noi ha his rsul mahs h rsul w found using h hang of as formula.

7 54 Chapr 4 Vido Eampl : nohr Eampl of Solving an Eponnial Equaion Eampl 3 In h firs sion, w prdid h populaion (in illions) of India yars afr 008 y using h funion f ( ).4( 0.034). If h populaion oninus following his rnd, whn will h populaion rah illion? W nd o solv for h so ha f() =.4(.034) Divid y.4 o isola h ponnial prssion Tak h arihm of oh sids of h quaion ln ln pply h ponn propry on h righ sid ln ln Divid oh sids y ln(.034) ln yars ln.034 If his growh ra oninus, h modl prdis h populaion of India will rah illion aou 4 yars afr 008, or approimaly in h yar 050. Try i Now 4. Solv 5(0.93) 0. In addiion o solving ponnial quaions, arihmi prssions ar ommon in many physial siuaions. Eampl 4 In hmisry, ph is a masur of h aidiy or asiiy of a liquid. Th ph is rlad o h onnraion of hydrogn ions, [H + ], masurd in mols pr lir, y h quaion ph H. If a liquid has onnraion of mols pr lir, drmin h ph. Drmin h hydrogn ion onnraion of a liquid wih ph of 7. To answr h firs qusion, w valua h prssion Whil w ould us our alulaors for his, w do no rally nd hm hr, sin w an us h invrs propry of s: ( 4) 4

8 Sion 4.3 Logarihmi Funions 55 To answr h sond qusion, w nd o solv h quaion 7 H. Bgin y isolaing h arihm on on sid of h quaion y muliplying oh sids y -: 7 H Rwriing ino ponnial form yilds h answr 7 H mols pr lir. Logarihms also provid us a mhanism for finding oninuous growh modls for ponnial growh givn wo daa poins. Vido Eampl : Convring an Eponnial Equaion from Coninuous Growh Form o nnual Growh Form Vido Eampl 3: Convring an Eponnial Equaion from nnual Growh Form o Coninuous Growh Form Eampl 5 populaion grows from 00 o 30 in wks. Find h oninuous growh ra. Masuring in wks, w ar looking for an quaion P() = 30. Using h firs pair of valus, r 0 00 a, so a = 00. P r ( ) a so ha P(0) = 00 and Using h sond pair of valus, r Divid y r Tak h naural of oh sids 00 r ln(.3) ln Us h invrs propry of s ln(.3) r ln(.3) r 0.3 This populaion is growing a a oninuous ra of 3.% pr wk.

9 56 Chapr 4 In gnral, w an rla h sandard form of an ponnial wih h oninuous growh form y noing (using k o rprsn h oninuous growh ra o avoid h onfusion of using r in wo diffrn ways in h sam formula): k a( r) a ( ) k r k r Using his, w s ha i is always possil o onvr from h oninuous growh form of an ponnial o h sandard form and vi vrsa. Rmmr ha h oninuous growh ra k rprsns h nominal growh ra for aouning for h ffs of oninuous ompounding, whil r rprsns h aual prn inras in on im uni (on wk, on yar,.). Eampl 6 ompany s sals an modld y h funion yars. Find h annual growh ra. S 0. ( ) 5000, wih masurd in k 0. Noing ha r, hn r 0. 75, so h annual growh ra is.75%. Th sals funion ould also wrin in h form S ( ) 5000( 0.75). Imporan Topis of his Sion Th Logarihmi funion as h invrs of h ponnial funion Wriing arihmi & ponnial prssions Propris of s Invrs propris Eponnial propris Chang of as Common Naural Solving ponnial quaions Try i Now nswrs ln( ) ln() ln(0.93). 4

10 Sion 4.3 Logarihmi Funions 57 Sion 4.3 Eriss Rwri ah quaion in ponnial form. 4( q) m. 3( ) k 3. ( a ) 4. ( z p ) u 5. v 6. r s 7. ln w n 8. ln y Rwri ah quaion in arihmi form y 0. 5 y a 4. 0 p v 5. d k k. h 6. z n y L Solv for ( ) 3 9. ( ) ( ) ln 4. ln Simplify ah prssion using arihm propris , ln ln Evalua using your alulaor ln ln 0.0 Solv ah quaion for h varial

11 58 Chapr 4 f a. Convr h quaion ino oninuous growh form, k 57. f f f f 400. f a. Convr h quaion ino annual growh form, f f f f Th populaion of Knya was 39.8 million in 009 and has n growing y aou.6% ah yar. If his rnd oninus, whn will h populaion d 45 million? 66. Th populaion of lgria was 34.9 million in 009 and has n growing y aou.5% ah yar. If his rnd oninus, whn will h populaion d 45 million? 67. Th populaion of Sal grw from 563,374 in 000 o 608,660 in 00. If h populaion oninus o grow ponnially a h sam ra, whn will h populaion d million popl? 68. Th mdian houshold inom (adjusd for inflaion) in Sal grw from $4,948 in 990 o $45,736 in 000. If i oninus o grow ponnially a h sam ra, whn will mdian inom d $50,000? 69. sinis gins wih 00 mg of a radioaiv susan. fr 4 hours, i has dayd o 80 mg. How long afr h pross gan will i ak o day o 5 mg? 70. sinis gins wih 00 mg of a radioaiv susan. fr 6 days, i has dayd o 60 mg. How long afr h pross gan will i ak o day o 0 mg? 7. If $000 is invsd in an aoun arning 3% ompoundd monhly, how long will i ak h aoun o grow in valu o $500? 7. If $000 is invsd in an aoun arning % ompoundd quarrly, how long will i ak h aoun o grow in valu o $300?