Instruction: Solving Exponential Equations without Logarithms. This lecture uses a four-step process to solve exponential equations:
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1 49 Instuction: Solving Eponntil Equtions without Logithms This lctu uss fou-stp pocss to solv ponntil qutions: Isolt th bs. Wit both sids of th qution s ponntil pssions with lik bss. St th ponnts qul to ch oth. 4 Solv fo th unknown. In shot, th sttgy involvs tnsfoming th qution to th fom b + f g. Consid th qution g f b thn solving. 8 Following th ubic, th bs should b isoltd. In oth wods, tnsfom th qution lgbiclly so tht th pssion + stnds lon on th lft sid of th qution s shown blow Now tht th bs is isoltd, th scond stp of th ubic quis tht th ight sid of th qution b wittn s n ponntil pssion with th sm bs s th lft sid (in this cs, th bs is two). + + Th substitution bov cn b pfomd bcus. Now, th thid stp of th ubic ugs stting th two ponnts, + nd, qul to on noth. Logic mintins tht th ponnts must b qul if th bss qul; thus, +. Finlly, th fouth stp of th ubic quis solving th sulting qution s blow. + 4 f g This sttgy only woks fo qutions tht cn b tnsfomd into th fom b b. Lt lctus will discuss ponntil qutions tht qui diffnt solution sttgis.
2 0 Instuction: Th Itionl Numb Th quntity dnotd by th symbol hs spcil significnc in mthmtics. It is n itionl numb tht is th bs of ntul logithms, nd it is oftn sn in l wold poblms tht involv ntul ponntil gowth o dcy. Th ppoimtd vlu of is Studnts cn plict this vlu by using th ^ ky on gphing clculto nd nting s th ponnt. Oftn nd ky must b usd to gnt ^. Th houstop symbol, ^, psnts th fct tht will b isd to som pow. Sinc is oftn involvd in ponntil gowth o dcy poblms, it is usully isd to som pow whn usd fo clcultions; thus, th clculto nticipts th nd to pply n ponnt to. Rising to th fist pow plicts th ppoimtd vlu of. Studnts cn lso gnt ppoimtions of by substituting tmly lg numbs fo m into th following pssion: m m + This pssion ppochs s th vibl m ppochs infinity. Functions of th fom y k o y k wh nd k l numb non-zo constnts oftn usd in scinc poblms. Th following poptis of ponnts cn b usd to simplify pssions with. ; ; s s s + ( ) ; ; 0 s s s ; nd + + Fo mpl, consid ( ) ( + + ) 9 6 ( + + )
3 Instuction: Gphing Eponntil Functions This lctu mploys stp-by-stp pocss fo gphing ponntil functions of th fom k y b. Th pocss pplis fo functions of this fom vn if th functions is tnsltd. Fo mpl, this discussion will consid k 4.. Idntify th hoizontl symptot, usully y 0. If th gph hs shift up o down c units, thn th qution y c dscibs th hoizontl symptot. Gph th hoizontl symptot ccodingly. Sinc th symptot is not ctully pt of th gph, us dottd lin to indict its position.. Find th y-intcpt by vluting th function whn th indpndnt vibl (th -vlu) is zo: k( ) 4 0 k(0) 4 k(0) 4 k(0) k(0) 7 k(0) 4 8
4 . If th function hs shift down, find th -intcpt. As discussd bov, finding th - intcpt quis solving th ponntil qution, which cn qui logithms. Fo k(), howv, logithms not ndd, which is somtims th cs s sn h Thus, solving th qution finds th -intcpt: k ( ) Plug fw -vlus into th function to find mo vlus of th function: k() Plot ths points nd sktch th cuv mmbing tht ponntil functions continully incs o continully dcs.
5 Instuction: Solving Eponntil Equtions without Logithms Empl Solving n Eponntil Equtions of th Fom + Solv th qution fo : g f b b Isolt th bs Rcogniz s th ninth pow of two. + 9 Not tht th two ponnts must b qul. + 9 If, thn 9 +. Solv th lin qution
6 4 Empl Solving n Eponntil Equtions of th Fom g f b b Solv th qution fo :. Not tht is th thid pow of. ( ) Rcll tht ( ) s s. Rcll tht s s fo ll non-zo vlus of. Not tht th two ponnts must b qul. If, thn. Solv th lin qution. 4
7 Instuction: Th Itionl Numb Empl Th Itionl Numb Evlut 7 7 without using clculto. Round th nsw to th nst thousndths. Rcll th ponnt popty of l numbs, s s + : Empl Th Itionl Numb Evlut 8 7 without using clculto. Round th nsw to th nst thousndths. Rcll th ponnt popty of non-zo l numbs, s s : Empl Th Itionl Numb ( Evlut ) without using clculto. Rcll th ponnt popty of l numbs, ( b s ) t t b s t. Rduc ppopitly nd cll th ponnt popty.
8 6 Empl 4 Th Itionl Numb ( Evlut ) 7 6. Round th nsw to th nst thousndth. Rcll th ponnt poptis of l numbs, s s + nd ( ) s s Rcll th ponnt popty of non-zo l numbs, s s. Us th clculto ky to vlut. Round to th nst thousndth Empl Th Itionl Numb 6 Simplify. Lv ll nsws ct. Rcll tht division with fctions is multipliction of th dividnd by th cipocl of th diviso Rcll th ponnt popty of non-zo l numbs, s s. 6 6
9 7 Empl 6 Th Itionl Numb Simplify + 4. Rcll th ponnt popty of l numbs, s s + nd simplify th dnominto. +. Fcto th gtst common fcto in th numto. Simplify nd duc. ( + ) ( + ) 4 + ( + 0 )
10 8 Gph y Instuction: Gphing Eponntil Functions Empl Gphing Eponntil Functions. Lbl intcpts nd symptots. Show pop bhvio. Th initil mount (th cofficint to th bs) is. Th y-intcpt is (0,). Th symptotic vlu is zo. Th common tio is. Sinc th common tio is gt thn on, th function incss nd ppochs zo s ppochs ngtiv infinity, mking th hoizontl symptot y 0 (th -is).
11 9 Gph q( ) Empl Gphing Eponntil Functions.. Lbl intcpts nd symptots. Show pop bhvio. Th initil mount (th cofficint to th bs) is. Th y-intcpt is (0,). Th symptotic vlu is zo. Th common tio is.. Sinc th common tio is gt thn on, th function incss nd ppochs zo s ppochs ngtiv infinity, mking th hoizontl symptot y 0 (th -is).
12 60 Empl Gphing Eponntil Functions Gph q( ) 0( 0.). Lbl intcpts nd symptots. Show pop bhvio. Th initil mount (th cofficint to th bs) is 0. Th y-intcpt is (0,0). Th symptotic vlu is zo. Th common tio is 0.. Sinc th common tio is lss thn on, th function dcss nd ppochs zo s ppochs infinity, mking th hoizontl symptot y 0 (th -is).
13 6 Empl 4 Gphing Eponntil Functions Gph g. Lbl intcpts nd symptots. Show pop bhvio. 4 Th initil mount (th cofficint to th bs) is. Th symptotic vlu is ngtiv two. Sinc +, th y-intcpt is (0, ). Th common tio is 4. Sinc th common tio is lss thn on, th function dcss nd ppochs ngtiv two (th symptotic vlu) s ppochs infinity, mking th hoizontl symptot y. Stting th qution qul to zo nd solving yilds th -vlu of th -intcpt (,0)
14 6 Suggstd Homwok fom Blitz Sction 4.: #7, #8, #, #7, #9, #4, #4, #47, # Appliction Ecis 0.00t In th ltt hlf of th 0th cntuy, Amicn sociologists usd p() t 86.9 to modl th pcnt of housholds tht w mid housholds btwn 970 nd 000 wh t psnts ys sinc 970. Accoding to th modl, wht pcnt of housholds w mid housholds in 990?
15 6 Pctic Poblms Solv th following qutions. # # # 8 # Hint: 7 8 # 6 #6 #7 6 #8, Simplify th pssions blow using poptis of ponnts. #9 #0 Gph th following functions. Lbl th intcpts. # t # f # y #4 h Fo fun, ct tbl of vlus nd gph th following spcil typ of ponntil function clld logistic gowth function o sigmoidl cuv. 00 # Q 0. ANSWERS # ½ # 4 # #7 / 4 #9 # # # 0,0 0, 0, 0 6 HA: y 00
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