湖北省武汉六中高三(国际部)数学复习课件：64对数与方程

Unit6:Exponential&LogarithmicEquations

Lesson3:SolvingLogarithmicEquationsLogarithmicEquationsAlogarithmicequationisanyequationinwhichthevariable–usuallyx–isinalogarithmInthislesson,we’lllookathowthesetypesofequationsaresolvedWhatYou’llNeedAgoodunderstandingofhowtoswitchbetweenlogarithmicandexponentialforms:

y=ax

logay=xTheloglaws:1.ZeroExponentloga1=02.ProductLawloga(xy)=logax+logay3.QuotientLaw4.PowerLawlogaxn=nlogax5.NegativeExponentslogax-n=–nlogaxWherea>0,a≠1,x>0,y>0Example1Solveforx:log(x+2)=log15Example1:Solution log(x+2)=log15 Becausewehaveasinglelogarithmontheleftandasinglelogarithmontherightandthebaseofthelogarithmsarethesame,wecandropthem: x+2=15 x=13Example1:NotesOnetechniqueusedtosolvelogarithmicequationsistowriteeachsideoftheequationasasinglelogarithmwithidenticalbasesOncethisisdone,youcansimplydropthebasesJustaswedidwithexponentialequationsinLesson6.1Example2Solveforw:log(w+4)=1Example2:SolutionSolve.Comparingthistothegenerallogarithmicform:logay=xSimplifyWehave: a=10 y=(w+4) x=1So,writingourequationinexponentialform(y=ax)gives:Example2:NotesAnotherusefultechniqueforsolvinglogarithmicequationsistoswitchfromlogarithmicformtoexponentialformDoingsogetsridofthelogarithm,allowingyoutoisolatexExample3Solveforx:Example3:SolutionWriteequationinexponentialformSimplifySolveforxExample4 Solveforx:log(x–1)–1=-log(x+2)Example4:SolutionMovethelogarithmstotheleftside,everythingelsetotherightApplyproductlawtocombinethetwologarithmsintoone(x–1)(x+2)=101Writetheequationinexponentialformx2+2x–x–2=10Expandandsimplifytheleft.Simplifytherightx2+x–2=10MoveeverythingtothelefttosolvethequadraticequationFactorthequadraticExample4:SolutionIfwereferbacktotheoriginalequation:Wecanseethatweneed:x>1andx>-2

youcanonlytakethelogarithmofnumberslargerthanzeroAsaresult,wemustrejectx=-4asapossiblesolution.Therefore,thesolutiontoisExample4:NotesAsthisexampleshows,youmustcombinethelogarithmsintoasinglelogarithmbeforeyoucanre-writetheequationinexponentialformThiswillrequiretheuseoftheloglawsInthisexample,wefoundthesolutionstobe:x=-4andx=3But,x=-4,turnstheleftsideoftheoriginalequationinto: log(-5)–1,whichisundefinedTherightsidebecamelog(-2),whichisalsoundefinedAsaresult,x=-4isnotasolutionRememberthatyoucan’ttakethelogarithmofzerooranegativeIfoneoftheanswersrequiresyoutotakethelogarithmofzerooranegative,youhaveto“throw”outthatanswerWhensolvinglogarithmicequationsalwayschecktomakesurethatyouranswersarevalidExample5Solveforx:

Example5:SolutionMoveallthelogarithmstotheleftApplythequotientlawWehaveasinglelogarithmontheleft–switchtoexponentialformMultiplybothsidesbyxtoclearthefractionSimplifytherightMoveeverythingtothelefttosolvethequadraticequationExample5:SolutionChecktoseeifthesesolutionsarevalidbypluggingthembackintotheoriginalequationSolution1:x=5FactorthequadraticExample5:SolutionBecauseweendupwithlog(-1),thesolutionx=-1isnotvalidTherefore,thesolutiontoisSolution2:x=-1SummaryTherearetwotechniquesyoucanusetosolvealogarithmicequation1.LikebasesWritebothsidesoftheequationassinglelogarithmswithidenticalbasesDropthebasesSolveforx2.ExponentialformMoveallthelogarithmstoonesideoftheequation,everythingelsetotheothersideUsetheloglawstocombinethelogarithmsintoasinglelogarithmRe-writetheequationinexponentialformSolveforxChecktomakesureyouranswersarevalidYoucan’ttakethelogarithmofzerooranegativeIfoneoftheanswersrequiresyoutotakethelogarithm