Mathematics. Exponentials and Logarithms. hsn.uk.net. Higher. Contents. Exponentials and Logarithms 134 HSN23300

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1 hs.uk.t Highr Mthmtics UNIT OUTCOME Epotils d Logrithms Cotts Epotils d Logrithms 4 Epotils 4 Logrithms Lws of Logrithms 4 Epotils d Logrithms to th Bs 9 Epotil d Logrithmic Equtios 4 Formul from Eprimtl Dt 4 7 Grph Trsformtios 4 HSN This documt ws producd spcill for th HSN.uk.t wbsit, d w rquir tht copis or drivtiv works ttribut th work to Highr Still Nots. For mor dtils bout th copright o ths ots, pls s

2 Highr Mthmtics Uit Mthmtics OUTCOME Epotils d Logrithms Epotils W hv lrd mt potil fuctios i Uit Outcom. Rcll tht fuctio i th form f ( ) is clld potil fuctio to th bs, whr >. If > th th grph looks lik this:, > O (, ) This is somtims clld growth fuctio If < < th th grph looks lik this:, < < O (, ) This is somtims clld dc fuctio Rmmbr tht th grph of potil fuctio f ( ) lws psss through (, ) d (, ) sic: f f hs.uk.t Pg 4 HSN

3 Highr Mthmtics Uit Mthmtics. Th ottr popultio o isld icrss b % pr r. How m full rs will it tk th popultio to doubl? Lt u b th iitil popultio. u u (% s dciml) u u u u u u u u u u For th popultio to doubl ftr rs, w rquir u u. So th cofficit of u, which is, must b t lst, i... Tr vlus of util this is stisfid (strtig with sic w kow < ): If, < If, < 4 If 4, 8 < If, > Thrfor ftr rs th popultio will doubl. O clcultor: ANS. Th fficic of mchi dcrss b % ch r. Wh th fficic drops blow 7%, th mchi ds to b srvicd. Aftr how m rs will th mchi d srvicd? Lt u b th iitil fficic. u 9 u (9% s dciml) u u u u u u u u u u Wh th fficic drops blow 7u (7% of th iitil vlu) th mchi must b srvicd. So th mchi ds srvicd ftr rs if 9 7. hs.uk.t Pg HSN

4 Highr Mthmtics Uit Mthmtics Tr vlus of util this is stisfid: If, 9 9 > 7 If, 9 87 > 7 4 If 4, 9 8 > 7 If, > 7 If, 9 7 < 7 Thrfor ftr rs, th mchi will hv to b srvicd. Logrithms Hvig prviousl dfid wht logrithm is (s Uit Outcom ) w wt to look i mor dtil t th proprtis of ths importt fuctios. Th rltioship btw logrithms d potils is prssd s: log whr, >. Hr, is th powr of which givs.. Writ i logrithmic form. log. Evlut log4. Th powr of 4 which givs is, so log 4 Lws of Logrithms Thr r thr lws of logrithms which ou must kow. Rul log + log log whr,, > If two logrithmic trms with th sm bs umbr ( bov) r big ddd togthr, th th trms c b combid b multiplig th rgumts ( d bov).. Simplif log + log4. log + log 4 log 4 log 8 hs.uk.t Pg HSN

5 Highr Mthmtics Uit Mthmtics Rul ( ) log log log whr,, > If logrithmic trm is big subtrctd from othr logrithmic trm with th sm bs umbr ( bov), th th trms c b combid b dividig th rgumts ( d i this cs). Not tht th rgumt which is big tk w ( bov) pprs o th bottom of th frctio wh th two trms r combid.. Evlut log 4 log4. Rul log log log4 log (Sic 4 4 ) log log whr, >. Th powr of th rgumt ( bov) c com to th frot of th trm s multiplir, d vic-vrs.. Eprss log7 i th form log7. log 7 log7 log 9 7 Sqush, Split d Fl Hr is simplr w to rmmbr th lws of logrithms: If logrithmic trms, with th sm bs umbr, r big ddd, th rgumts r squshd togthr b multiplig thm If logrithmic trm is big subtrctd from othr, with th sm bs umbr, th rgumts r split ito frctio Th powr of rgumt c fl to th frot of th log trm d vicvrs. hs.uk.t Pg 7 HSN

6 Highr Mthmtics Uit Mthmtics Not Wh workig with logrithms, th followig fcts r usful: log sic 4. Evlut log7 7 + log. log 7 + log + 7 Combiig svrl log trms log sic Wh ddig d subtrctig svrl log trms i th form log b, thr is simpl w to combi ll th trms i o stp. Multipl th rgumts of th positiv log trms i th umrtor, Multipl th rgumts of th gtiv log trms i th domitor.. Evlut log + log log log + log log log log. Evlut log 4 + log. log 4 + log log 4 + log log 4 + log 9 log 4 9 log log (Sic ) rgumts of positiv log trms rgumts of gtiv log trms log OR log 4 + log log + log log + log ( ) ( ) log + log log log + log + log log (Sic log ) hs.uk.t Pg 8 HSN

7 Highr Mthmtics Uit Mthmtics 4 Epotils d Logrithms to th Bs Th costt is importt umbr i Mthmtics, prticulrl wh dlig with rl-lif situtios. Its vlu is roughl (to 9 d.p.), d is dfid s: + s If ou tr vr lrg vlus of o our clcultor, ou will gt clos to th vlu of. Lik π, is irrtiol umbr. Throughout this sctio, w will us i prssios of th form:, which is clld potil to th bs log, which is clld logrithm to th bs. This is lso kow s th turl logrithm of, d is oft writt s l (i.. l log ).. Writ dow th vlu of log 8 log 8. 8 (to d.p.). Solv log 9. log 9 so (to d.p.).. Simplif 4log ( ) log ( ) prssig our swr i th form + log b log c whr, b d c r whol umbrs. 4log ( ) log ( ) 4 log + 4 log log log 4 log + 4 log + 4 log log + log log 4 + log log 7 O clcultor: l 8 O clcultor: 9 OR 4log ( ) log ( ) log ( ) log ( ) 4 ( ) log ( ) 4 4 Rmmbr log 7 b b log 7 log + log log 7 + log log 7 hs.uk.t Pg 9 HSN

8 Highr Mthmtics Uit Mthmtics Epotil d Logrithmic Equtios M mthmticl modls of rl-lif situtios us potils d logrithms. It is importt to bcom fmilir with usig th lws of logrithms to hlp solv qutios.. Solv log + log log 7 for >. log + log log 7 log log 7 7 (Sic log log ). Solv ( ) ( ) log 4 + log for >. ( ) ( ) log 4 + log 4 log (Sic log ) 4 + ( ) log p + + log p log p for p > 4.. Solv log ( p + ) + log ( p ) log ( p) log ( p + )( p ) log ( p) ( p + )( p ) p p p + p p p + p 8 p ( p )( p ) + or p p p Sic w rquir p > 4, p is th solutio. hs.uk.t Pg 4 HSN

9 Highr Mthmtics Uit Mthmtics Dlig with Costts Somtims it m b cssr to writ costts s logs, i ordr to solv qutios. 4. Solv log 7 log + for >. Writ i logrithmic form: log (Sic log ) log log 8 Us this i th qutio: log 7 log + log 8 log 7 log OR log 7 log + log 7 log 7 log Covrtig from log to potil form: Solvig Equtios with ukow Epots If ukow vlu (.g. ) is th powr of trm (.g. or ), d its vlu is to b clcultd, th w must tk logs o both sids of th qutio to llow it to b solvd. Th sm solutio will b rchd usig bs, but clcultors c b usd for vlutig logs to th bs d.. Solv 7. Tkig log of both sids log log 7 log log 7 ( log ) log 7 94 (to d.p.) OR Tkig log of both sids log log 7 log log 7 log 7 log 94 (to d.p.) hs.uk.t Pg 4 HSN

10 Highr Mthmtics Uit Mthmtics. Solv log log 4 ( + ) log log 4 log 4 + log (to d.p.) OR + log log 4 ( + ) log log 4 log 4 + log (to d.p.) t 7. For th formul s ( t ) : () Evlut s. (b) Clcult th vlu of t whr s ( t ) s. () s (Sic ) (b) Clcult th vlu of t whr s ( t ) s : log t t t log t log log ( ) t log (Sic log ) t (to d.p.) Formul from Eprimtl Dt Rsults from primts oft rlt to potil fuctios, ithr of th form b or b. It is oft mor usful to hv this iformtio s stright li grph, bcus it is th sir to fid its qutio. To obti stright li, th qutios must b writt i th form m + c. Th vlus of d b c th b dtrmid from this qutio. hs.uk.t Pg 4 HSN

11 Highr Mthmtics Uit Mthmtics Equtios i th form b. Th rsults from primt wr otd s follows: Th rltioship btw ths dt c b writt i th form. Fid th vlus of d b, d stt th formul for i trms of. W d to prss th qutio i th form m + c : b log log (tkig logs of both sids) log log log log + log log log + b log Compr to m + c : b i.. log b log + log b W c fid th grdit b, usig two poits o th li: Usig ( 7, ) d ( 8, ), b (to d.p.) log 9 log + log. So Now w c work out b substitutig poit ito this qutio: Usig ( 7, ), log log (to d.p.) b Thrfor 9 4. Not Dpdig o th poits usd, slightl diffrt swrs m b obtid. hs.uk.t Pg 4 HSN

12 Highr Mthmtics Uit Mthmtics Equtios i th form b. Th rsults from primt wr otd s follows: Th rltioship btw ths dt c b writt i th form b. Fid th vlus of d b, d stt th formul for i trms of. W d to prss th qutio i th form m + c : log log b (tkig logs of both sids) b log log + log b log log + log b Compr to m + c : i.. log log b + log W c fid th grdit log b (d hc b), usig two poits o th li: Usig (, 4 ) d ( 8, 4 ), log b So log 7 + log. So b (to d.p.) 7 8 (to d.p.) Now w c work out log (d hc ) b substitutig poit ito this qutio: Usig (, 4 ), Thrfor log log 4 d so log log 4 7 so 9 (to d.p.) 9 97 (to d.p.) hs.uk.t Pg 44 HSN

13 Highr Mthmtics Uit Mthmtics 7 Grph Trsformtios Grph trsformtios wr covrd i Uit Outcom Fuctios d Grphs, but w c ow look i mor dtil t pplig trsformtios to grphs of potil d logrithmic fuctios.. Show blow is th grph of f ( ) whr f ( ) log. f ( ) O ( 9,) () Stt th vlu of. (b) Sktch th grph of f ( + ) +. () log 9 (sic 9) (b) Th grph shifts two uits to th lft, d o uit upwrds: ( 7,) O f ( + ) +. Show blow is prt of th grph of log. log O (,) Sktch th grph of log ( ) log ( ) log log So rflct i th -is. O log (, ) ( ) hs.uk.t Pg 4 HSN