MATHEMATICS FOR MANAGEMENT BBMP1103
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1 Objctivs: TOPIC : EXPONENTIAL AND LOGARITHM FUNCTIONS. Idntif pnntils nd lgrithmic functins. Idntif th grph f n pnntil nd lgrithmic functins. Clcult qutins using prprtis f pnntils. Clcult qutins using prprtis f lgrithms 5. Slv pplictin prblms INTRODUCTION. Bth pnntil nd lgrithm functin r n-t-n functins.. Th invrs functin f n pnntil functin is clld th lgrithm functin.. PROPERTIES OF EXPONENTIALS. A functin is clld n pnntil functin if it hs frm whr th bs is psitiv, with nd tht its pnnt is n rl numbr. () b b () () () b () (7) () (5) (9) Empls : Find th vlus f () (b) (c) P g Prprd b Ezidin bin Nrmn Tutr
2 P g Prprd b Ezidin bin Nrmn Tutr (d) () (f) Slutins : () (b) (c) (d) 9 () (f) 9 9 Empls : Slv () (b) (c) (d) 9 Slutins : () (b)
3 (c) (d) 9, Ercis. Find th vlus f () (b) (c) 7 (d) () 5 (f) Ercis. () (b) (c) (d) () (f) P g Prprd b Ezidin bin Nrmn Tutr
4 . EQUATIONS AND GRAPHS SKETCHING. Thr r tw gnrl shps f pnntils grphs.. Th shp dpnd n th bs vlu f th pnntil functins. (i) whr > (ii) whr < < Th fllwing r th prprtis f th grph f pnntil functin f. (i) Th dmin fr pnntil functins is th ntir rl numbrs (ii) Its rng is ll psitiv numbrs (iii) Th intrcpt n th pnntil grph is,. (iv) Thr is n intrcpt (vi) If whr >, th grph is incrsing frm lft t right (vii) If whr < <, th grph is dcrsing frm lft t right P g Prprd b Ezidin bin Nrmn Tutr
5 Empl : Sktch grph f Slutin :. Stp (Cnstruct tbl cnsisting svrl vlus f nd.) Stp (Plt th pints n pln.) Stp (Drw smth curv thrugh ll th plttd pints.) P g 5 Prprd b Ezidin bin Nrmn Tutr
6 Empl : Sktch grph f. Slutin : Stp (Cnstruct tbl cnsisting svrl vlus f nd.) Stp (Plt th pints n pln.) Stp (Drw smth curv thrugh ll th plttd pints.) P g Prprd b Ezidin bin Nrmn Tutr
7 . LOGARITHM FUNCTIONS. A lgrithm functin with bs, is wittn s lg whr >,.. is th lgrithm fr with bs, dntd b lg.. Lgrithm with bs f is knwn s cmmn lgrithm nd is writtn s lg lg lg.. Lgrithm with bs, is clld nturl lgrithm, dntd b lg ln. lg Lgrithm Frm Epnntil Frm Empl : Cnvrt th fllwing qutins, frm lgrithm t pnntil frms. () lg 9 (b) lg (c) lg Slutin : () 9 (b) (c) Empl : Cnvrt th fllwing qutins, frm pnntil t lgrithm frms. () 5 (b) (c) 5 Slutin : () lg 5 (b) lg (c) lg 5 P g 7 Prprd b Ezidin bin Nrmn Tutr
8 . PROPERTIES OF LOGARITHMS () lg () lg m lg m () lg b m lg m (Lgrithm bs intrchngbl frmul lg () lg M lg N lg MN M (5) lg M lg N lg N () lg M lg N thn M N b Empls : Using th bv prprtis, find th vlu fr: () lg (b) ln (c) lg (d) lg () lg lg (f) lg 5 lg 9 Slutins: () lg lg lg (b) ln lg lg (c) lg lg lg (d) lg lg lg lg () lg lg lg lg lg (f) lg 5 lg 9 5 lg 9 lg P g Prprd b Ezidin bin Nrmn Tutr
9 Empls : Find th vlu f. () lg lg (b) lg (c) lg (d) lg () lg lg lg lg Slutin : (f) lg () lg lg 5 (c) lg 9 () lg lg lg lg (b) lg, (d) lg lg lg lg lg (f) lg lg lg P g 9 Prprd b Ezidin bin Nrmn Tutr
10 .5 EQUATIONS AND GRAPH SKETCHING. Thr r tw gnrl shps f lgrithm grphs. Th dpnd vr much n th bs vlu f lgrithm funfins. (i) lg whr > (ii) lg whr < < Th fllwing r th prprtis f th grph f pnntil functin f lg. (i) Th dmin fr lgrithm functins is ll psitiv numbrs (ii) Its rng is th ntir rl numbrs (iii) Th intrcpt n th lgrithm grph is,. (iv) Thr is n intrcpt (viii) If >, th grph is incrsing frm lft t right (i) If < <, th grph is dcrsing frm lft t right P g Prprd b Ezidin bin Nrmn Tutr
11 Empl : Sktch grph f lg. Slutin : Stp (Cnstruct th qutin, frm lgrithm t pnntil frm). lg Stp (Cnstruct tbl cnsisting svrl vlus f nd.) Stp (Plt th pints n pln.) Stp (Drw smth curv thrugh ll th plttd pints.) P g Prprd b Ezidin bin Nrmn Tutr
12 Empl : Sktch grph f lg. Slutin : Stp (Cnstruct th qutin, frm lgrithm t pnntil frm). lg Stp (Cnstruct tbl cnsisting svrl vlus f nd.) Stp (Plt th pints n pln.) Stp (Drw smth curv thrugh ll th plttd pints.) P g Prprd b Ezidin bin Nrmn Tutr
13 .5. APPLICATION ON GROWTH AND DECAY PROCESSES. Epnntil functin cn b pplid int grwth nd dc prcsss.. Th frmul fr ttl grwth is P P rt Whr P P r t = numbr f rsidnts ftr rs = numbr f riginl rsidnts = prcntg (rt) f grwth = tim prid Empl: Supps th ttl numbr f rsidnts in givn twn is, nd th rt f grwth f th rsidnts is 5% pr r. () Dtrmin th ttl numbr f rsidnts in this twn in th prid f rs frm nw. (b) Hw mn rs will it tk fr th numbr f rsidnts t dubl? Slutin: () Substitut ll th givn vlus int th frmul t find th vlu f P? P P rt P, r 5% t rt P P Th numbr f th rsidnts ftr si mr rs is 997. (b) Dubling th numbr f rsidnts implis P P. P g Prprd b Ezidin bin Nrmn Tutr
14 P P lg P P.5t lg rt.5t ln.5t.5t ln t.5 t. Th numbr f th rsidnts will dubl in but rs.. Th frmul fr dc prcss is P P rt Empl: Supps rdictiv lmnt is ging thrugh pwr dc ftr t ds bsd n pnntil t functin P.75. Hw much f th quntit is lft ftr ds? Slutin: Substitut ll th givn vlus int th frmul t find th vlu f P? P P rt INVESTMENT WITH COMPOUND INTEREST Th ttl munt f mn, dntd b S is th cmpund munt fr sum f mn P cmpunding ftr n th r, whr th intrst is pbl k tims t th rt f pr r % nnum, is givn b th frmul blw: S P r k nk Whr P g Prprd b Ezidin bin Nrmn Tutr
15 S P r k k = cmpund munt r th prspctiv vlu = initil invstmnt r th principl vlu = intrst rt pr nnum = numbr f intrst pid (cmpund) in r = numbr f r Empl : If RM is invstd t th rt f % pr nnum, cmpunding (pbl) vr qurtrl, wht wuld th ttl munt b in th ccunt ftr rs? Slutin: S?, P, r %, r, n S P. S r k S.5 S. S. nk Empl : Dtrmin th principl munt f ln, givn tht th prspctiv munt pbl ftr rs is RM,59. nd th cmpund rt f % pr nnum, cmpunding (pbl) n rl bsis. Slutin: S 59., P?, r %, r, n S P. 59. P 59. P. 59. P.59 P P r k nk P g 5 Prprd b Ezidin bin Nrmn Tutr
16 Ercis.. () Givn tht th pric n cr f lnd is incrsing t th rt f % pr r. Hw lng will it tk fr th pric t incrs t RM,, givn its currnt vlu is RM,. (b) Du t cnm dwnfll, th ttl numbr f rsidnts in twnship is rducing t th rt f % pr r. Initil ppultin ws, rsidnts. Wht is th ppultin ftr r?. Dtrmin th cmpund munt, givn th principl vlus, cmpund intrst rts nd tim prids: () RM55; % pr nnum cmpunding n mnthl bsis; mnths. (b) RM,; % pr nnum cmpunding rl; 5 rs (c) RM7; 7.% pr nnum cmpunding n qurtrl bsis; 5 rs mnths. (d) RM; 5.75% pr nnum cmpunding dil; 5 ds (ssum r = 5 ds).. Dtrmin th principl munt, givn th fllwing cmpund vlus, cmpund intrst rts nd tim prids: () RM,.; % pr nnum cmpunding n mnthl bsis; mnths (b) RM,97.; 5.% pr nnum cmpunding dil; 5 ds (ssum r = 5 ds) (c) RM57.;.% pr nnum cmpunding vr mnths; mnths (d) RM,.; 7.% pr nnum cmpunding vr mnths; 5 rs mnths P g Prprd b Ezidin bin Nrmn Tutr
17 P g 7 Prprd b Ezidin bin Nrmn Tutr
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