Algebra 2 Unit 4B: Logarithms - Ms. Talhami · Algebra 2 Unit 4B: Logarithms Ms. Talhami 3 Exercise #2: Evaluate the following logarithms. If needed, write an equivalent exponential

Algebra2Unit4B:Logarithms

Ms.Talhami 1


Name_________________


Ms.Talhami 2

INTRODUCTION TO LOGARITHMS COMMONCOREALGEBRAII

Exponentialfunctionsareofsuchimportancetomathematicsthattheirinverses,functionsthat“reverse”theiraction,areimportantthemselves.Thesefunctions,knownaslogarithms,willbeintroducedinthislesson.Exercise#1:Thefunction ( ) 2xf x = isshowngraphedontheaxesbelowalongwithitstableofvalues.(a)Isthisfunctionone-to-one?Explainyouranswer.(b) Basedonyouranswerfrompart(a),whatmustbe

trueabouttheinverseofthisfunction?(c) Create a table of values below for the inverse of

( ) 2xf x = andplotthisgraphontheaxesgiven.(d)Whatwouldbethefirststeptofindanequationforthisinversealgebraically?Writethisstepdownand

thenstop.Defining Logarithmic Functions – The function logby x= is the namewe give the inverse of xy b= . Forexample, 2logy x= istheinverseof 2xy = .BasedonExercise#1(d),wecanwriteanequivalentexponentialequationforeachlogarithmasfollows:

log is the same as yby x b x= =

Basedonthis,weseethatalogarithmgivesasitsoutput(y-value)theexponentwemustraisebtoinordertoproduceitsinput(x-value).

x 0 1 2 3

1 2 4 8

x

y

x

Notice that, as always, thegraphs of and

aresymmetricacross


Ms.Talhami 3

Exercise#2: Evaluatethefollowinglogarithms. Ifneeded,writeanequivalentexponentialequation. Doasmanyaspossiblewithouttheuseofyourcalculator.

(a) 2log 8 (b) 4log 16 (c) 5log 625 (d) 10log 100,000 (e) ( )6

1log 36 (f) ( )21log 16 (g) 5log 5 (h) 5

3log 9

Itiscriticallyimportanttounderstandthatlogarithmsgiveexponentsastheiroutputs.Wewillbeworkingformultiplelessonsonlogarithmsandabasicunderstandingoftheirinputsandoutputsiscritical.Exercise#3: If thefunction ( )2log 8 9y x= + + wasgraphed inthecoordinateplane,whichofthefollowingwouldrepresentitsy-intercept? (1)12 (3)8 (2)13 (4)9Exercise#4:Betweenwhichtwoconsecutiveintegersmust 3log 40 lie? (1)1and2 (3)3and4 (2)2and3 (4)4and5CalculatorUseandLogarithms–Mostcalculatorsonlyhavetwologarithmsthattheycanevaluatedirectly.Oneofthem, 10log x ,issocommonthatitisactuallycalledthecommonlogandtypicallyiswrittenwithoutthebase10.

10log logx x= (TheCommonLog)

Exercise#5:Evaluateeachofthefollowingusingyourcalculator.

(a) log100 (b) ( )1log 1000 (c) log 10


Ms.Talhami 4

INTRODUCTION TO LOGARITHMS COMMONCOREALGEBRAIIHOMEWORK

FLUENCY

1. Whichofthefollowingisequivalentto 7logy x= ?

(1) 7y x= (3) 7 yx =

(2) 7x y= (4)17y x=

2. Ifthegraphof 6xy = isreflectedacrosstheline y x= thentheresultingcurvehasanequationof

(1) 6xy = − (3) 6logx y=

(2) 6logy x= (4) 6x y= 3. Thevalueof 5log 167 isclosesttowhichofthefollowing?Hint–guessandchecktheanswers.

(1)2.67 (3)4.58

(2)1.98 (4)3.184. Whichofthefollowingrepresentsthey-interceptofthefunction ( )log 1000 8y x= + − ?

(1) 8− (3)3

(2) 5− (4)55. Determinethevalueforeachofthefollowinglogarithms.(Easy) (a) 2log 32 (b) 7log 49 (c) 3log 6561 (d) 4log 10246. Determinethevalueforeachofthefollowinglogarithms.(Medium) (a) ( )2

1log 64 (b) ( )3log 1 (c) ( )51log 25 (d) ( )7

1log 343


Ms.Talhami 5

7. Determinethevalueforeachofthefollowinglogarithms.Eachofthesewillhavenon-integer,fractionalanswers.(Difficult)

(a) 4log 2 (b) 4log 8 (c) 35log 5 (d) 5

2log 4 8. Betweenwhattwoconsecutiveintegersmustthevalueof 4log 7342 lie?Justifyyouranswer.9. Betweenwhattwoconsecutiveintegersmustthevalueof ( )5

1log 500 lie?Justifyyouranswer.

APPLICATIONS10.In chemistry, the pH of a solution is defined by the equation ( )pH log H= − whereH represents the

concentrationofhydrogenionsinthesolution.AnysolutionwithapHlessthan7isconsideredacidicandanysolutionwithapHgreaterthan7isconsideredbasic.Fillinthetablebelow.RoundyourpH’stothenearesttenthofaunit.

REASONING

11.Canthevalueof ( )2log 4− befound?Whataboutthevalueof 2log 0?Whyorwhynot?Whatdoesthistellyouaboutthedomainof logb x ?

Substance ConcentrationofHydrogen pH Basicor

Acidic?

Milk

Coffee

Bleach

LemonJuice

Rain


Ms.Talhami 6

GRAPHS OF LOGARITHMS COMMONCOREALGEBRAII

Thevastmajorityoflogarithmsthatareusedintherealworldhavebasesgreaterthanone;thepHscalethatwesawonthelasthomeworkassignmentisagoodexample.Inthislessonwewillfurtherexploregraphsoftheselogarithms,includingtheirconstruction,transformations,anddomainsandranges.Exercise#1:Considerthelogarithmicfunction 3logy x= anditsinverse 3xy = .(a) Constructatableofvaluesfor 3xy = andthenusethisto

constructatableofvaluesforthefunction 3logy x= .(b) Graph 3xy = and 3logy x= onthegridgiven.Labelwithequations.(c)Statethenaturaldomainandrangeof 33 and logxy y x= = .Exercise#2:Usingyourcalculator,sketchthegraphof 10logy x= ontheaxesbelow.Labelthex-intercept.Statethedomainandrangeof 10logy x= .

Domain:

Range:

Domain:Range:

Domain:Range:

y

x

2

10

y

x


Ms.Talhami 7

Exercise#3: Whichofthefollowingequationsdescribesthegraphshownbelow?Showorexplainhowyoumadeyourchoice. (1) ( )3log 2 1y x= + − (2) ( )2log 3 1y x= − + (3) ( )2log 3 1y x= + − (4) ( )3log 3 1y x= + − Thefactthatfindingthelogarithmofanon-positivenumber(negativeorzero)isnotpossibleintherealnumbersystemallowsustofindthedomainsofavarietyoflogarithmicfunctions.Exercise#4:Determinethedomainofthefunction ( )2log 3 4y x= − .Stateyouranswerinset-buildernotation.Alllogarithmswithbaseslargerthan1arealwaysincreasing.Thisincreasingnaturecanbeseenbycalculatingtheiraveragerateofchange.Exercise#5:Considerthecommonlog,orlogbase10, ( ) ( )logf x x= .(a) Set up and evaluate an expression for the average rate of

changeof ( )f x overtheinterval1 10x≤ ≤ (b) Set up and evaluate an expression for the average rate of

changeof ( )f x overtheinterval1 100x≤ ≤ .(c) Whatdothesetwoanswerstellyouaboutthechangingslopeofthisfunction?

y

x


Ms.Talhami 8

GRAPHS OF LOGARITHMS COMMONCOREALGEBRAIIHOMEWORK

FLUENCY

1. Thedomainof ( )3log 5y x= + intherealnumbersis

(1){ }| 0x x > (3){ }| 5x x > (2){ }| 5x x > − (4){ }| 4x x ≥ − 2. Whichofthefollowingequationsdescribesthegraphshownbelow? (1) 5logy x= (3) 3logy x= (2) 2logy x= (4) 4logy x= 3. Whichofthefollowingrepresentsthey-interceptofthefunction ( )2log 32 1y x= − − ? (1)8 (3) 1− (2) 4− (4)44. Whichofthefollowingvaluesofxisnotinthedomainof ( ) ( )5log 10 2f x x= − ? (1) 3− (3)5 (2)0 (4) 4 5. Whichofthefollowingistrueaboutthefunction ( )4log 16 1y x= + − ? (1)Ithasanx-interceptof4anday-interceptof 1− . (2)Ithasx-interceptof 12− anday-interceptof1. (3)Ithasanx-interceptof 16− anday-interceptof1. (4)Ithasanx-interceptof 16− anday-interceptof 1− .

y

x


Ms.Talhami 9

6. Determine the domains of each of the following logarithmic functions. State your answers using anyacceptednotation.Besuretoshowtheinequalitythatyouaresolvingtofindthedomainandtheworkyouusetosolvetheinequality.

(a) ( )5log 2 1y x= − (b) ( )log 6y x= − 7. Graphthelogarithmicfunction 4logy x= onthegraphpapergiven.Foramethod,seeExercise#1.REASONING8. Logarithmic functions whose bases are larger than 1 tend to increase very slowly as x increases. Let's

investigatethisfor ( ) ( )2logf x x= . (a)Findthevalueof ( ) ( ) ( ) ( )1 , 2 , 4 , and 8f f f f withoutyourcalculator. (b) Forwhatvalueofxwill ( )2log 10x = ?Forwhatvalueofxwill ( )2log 20x = ?

y

x


Ms.Talhami 10

LOGARITHM LAWS COMMONCOREALGEBRAII

Logarithmshaveproperties,justasexponentsdo,thatareimportanttolearnbecausetheyallowustosolveavarietyofproblemswherelogarithmsareinvolved.Keepinmindthatsincelogarithmsgiveexponents,thelawsthatgovernthemshouldbesimilartothosethatgovernexponents.Belowisasummaryoftheselaws.Exercise#1:Whichofthefollowingisequalto ( )3log 9x ?

(1) 3 3log 2 log x+ (3) 32 log x+

(2) 32log x (4) 3log 2x +

Exercise#2:Theexpression ( )2log 1000

x canbewritteninequivalentformas

(1)2log 3x − (3)2log 6x −

(2) log 2 3x − (4) log 2 6x − Exercise#3: If log3 and log 2a b= = thenwhichof the followingcorrectlyexpresses thevalueof log12 intermsofaandb? (1) 2a b+ (3) 2a b+

(2) 2a b+ (4) 2a b+

Exercise#4:Whichofthefollowingisequivalentto 52log xy

⎛ ⎞⎜ ⎟⎝ ⎠

?

(1) 2 2log 5logx y− (3) 2 21 log 5log2

x y−

(2) 2 22log 5logx y+ (4) 2 22log 5logx y−


Ms.Talhami 11

Exercise#5:Thevalueof 35log 27

⎛ ⎞⎜ ⎟⎝ ⎠

isequalto

(1) 3log 5 62− (3) 3log 5 3

2−

(2) 32log 5 3+ (4) 32log 5 3−

Exercise#6:If ( ) ( )logf x x= and ( ) 3100g x x= then ( )( )f g x = (1)100log x (3)300log x (2)6 log x+ (4)2 3log x+

Exercise#7:Thelogarithmicexpression 72log 32x canberewrittenas

(1) 2log 35x (3) 25 7 log x+

(2) 25 7 log2

x+ (4) 235 log2

x+

Exercise#8:If log7 k= then ( )log 4900 canbewrittenintermsofkas (1) ( )2 1k + (3) ( )2 3k − (2) 2 1k − (4) 2 1k + The logarithm laws are important for future study inmathematics and science. Being fluent with them isessential.Arguably,themostimportantofthethreelawsisthepowerlaw.Inthenextexercise,wewillexamineitmoreclosely.Exercise#9:Considertheexpression ( )2log 8x .

(c) Showthat ( )2log 8 3x x= byrewriting 38 as 2 .

(a) Usingthethirdlogarithmlaw(theProductLaw),rewritethisasequivalentproductandsimplify.

(b) Test the equivalency of these two expressionsfor .


Ms.Talhami 12

LOGARITHM LAWS COMMONCOREALGEBRAIIHOMEWORK

FLUENCY

1. Whichofthefollowingisnotequivalentto log36? (1) log 2 log18+ (3) log30 log6+ (2)2log6 (4) log 4 log9+ 2. The 3log 20 canbewrittenas (1) 3 32log 2 log 5+ (3) 3 3log 15 log 5+ (2) 32log 10 (4) 3 32log 4 3log 4+

3. Whichofthefollowingisequivalentto3

3log x

y⎛ ⎞⎜ ⎟⎝ ⎠

?

(1) log logx y− (3) 13log log3

x y−

(2) ( )9log x y− (4) ( ) ( )log 3 log 3yx −

4. Thedifference ( ) ( )2 2log 3 log 12− isequalto

(1) 2− (3) 14

(2) 12

− (4) 4

5. If log5 and log 2 then log 200p q= = canbewrittenintermsofpandqas (1) 4p q+ (3) ( )2 p q+ (2) 2 3p q+ (4)3 2p q+


Ms.Talhami 13

6. Whenroundedtothenearesthundredth, 3log 7 1.77= .Whichofthefollowingrepresentsthevalueof

3log 63 tothenearesthundredth?Hint:write63asaproductinvolving7. (1)3.54 (3)3.77 (2)8.77 (4)15.93

7. Theexpression 14log log 3log2

x y z− + canberewrittenequivalentlyas

(1)4 3

log x zy

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

(3)4 3

log2x zy

⎛ ⎞⎜ ⎟⎝ ⎠

(2) 6log xzy

⎛ ⎞⎜ ⎟⎝ ⎠

(4)4 36log x zy

⎛ ⎞⎜ ⎟⎝ ⎠

8. If 2log 3k = then 2log 48 = (1) 2 3k + (3) 8k + (2)3 1k + (4) 4k + 9. If ( ) ( ) ( ) ( )( )6

48 and log 2 then g x x f x x f g x= = =? (1) 44log 1x + (3) ( )42 3log 1x+ (2) ( )43 log 2x+ (4) 46log 4x + REASONING10.Considertheexponentialequation4 30x = .

(c) The solution to the original equation is ( )( )

log 30log 4

x = , can you see why based on (b)? Evaluate this

expressionandchecktoseeitiscorrect.

(a) Betweenwhattwoconsecutiveintegersmustthesolutiontothisequationlie?Explainyourreasoning.

(b)Write asanequivalentproductusingthethirdlogarithmlaw.


Ms.Talhami 14

SOLVING EXPONENTIAL EQUATIONS USING LOGARITHMS COMMONCOREALGEBRAII

Earlierinthisunit,weusedtheMethodofCommonBasestosolveexponentialequations.Thistechniqueisquitelimited,however,becauseitrequiresthetwosidesoftheequationtobeexpressedusingthesamebase.Amoregeneralmethodutilizesourcalculatorsandthethirdlogarithmlaw:

Exercise#1:Solve:4 8x = using(a)commonbasesand(b)thelogarithmlawshownabove.

(a)MethodofCommonBases (b)LogarithmApproachThebeautyofthislogarithmlawisthatitremovesthevariablefromtheexponent.Thislaw,incombinationwiththelogarithmbase10,thecommonlog,allowsustosolvealmostanyexponentialequationusingcalculatortechnology.Exercise#2: Solveeachof the followingequations for the valueofx. Roundyouranswers to thenearesthundredth.

(a)5 18x = (b) 4 100x = (c) 2 1560x = Theseequations canbecomemore complicated,buteachandevery timewewill use the logarithm law totransformanexponentialequationintoonethatismorefamiliar(linearonlyfornow)Exercise#3:Solveeachofthefollowingequationsforx.Roundyouranswerstothenearesthundredth.

(a) 36 50x+ = (b) ( ) 521.03 2x− =

THETHIRDLOGARITHMLAW


Ms.Talhami 15

Nowthatwearefamiliarwiththismethod,wecanrevisitsomeofourexponentialmodelsfromearlierintheunit.Recallthatforanexponentialfunctionthatisgrowing:

Exercise#4:Abiologistismodelingthepopulationofbatsonatropicalisland.Whenhefirststartsobservingthem,thereare104bats.Thebiologistbelievesthatthebatpopulationisgrowingatarateof3%peryear.

Exercise#5:Astockhasbeendeclininginpriceatasteadypaceof5%perweek.Ifthestockstartedatapriceof$22.50pershare,determinealgebraicallythenumberofweeks itwill takeforthepricetoreach$10.00.Roundyouranswertothenearestweek.Asafinaldiscussion,wereturntoevaluatinglogarithmsusingourcalculator.Manymoderncalculatorscanfindalogarithmofanybase.Somestillonlyhavethecommonlog(base10)andanotherthatwewillsoonsee.But,wecanstillexpressouranswersintermsoflogarithms.

Exercise#6:Findthesolutiontoeachofthefollowingexponentialequationsintermsofalogarithmwiththesamebaseastheexponentialequation.

IfquantityQisknowntoincreasebyafixedpercentagep,indecimalform,thenQcanbemodeledby

where representstheamountofQpresentat andtrepresentstime.

(a) Writeanequationforthenumberofbats,,asa functionofthenumberofyears,t,sincethebiologiststartedobservingthem.

(b)Usingyourequationfrompart(a),algebraicallydetermine thenumber of years itwill take forthe bat population to reach 200. Round youranswertothenearestyear.

(a) (b)


Ms.Talhami 16

SOLVING EXPONENTIAL EQUATIONS USING LOGARITHMS COMMONCOREALGEBRAIIHOMEWORK

FLUENCY

1. Whichofthefollowingvalues,tothenearesthundredth,solves:7 500x = ?

(1)3.19 (3)2.74

(2)3.83 (4)2.17

2. Thesolutionto 32 52x= ,tothenearesttenth,iswhichofthefollowing?

(1)7.3 (3)11.4 (2)9.1 (4)17.13. Tothenearesthundredth,thevalueofxthatsolves 45 275x− = is (1)6.73 (3)8.17 (2)5.74 (4)7.49

4. Solveeachofthefollowingexponentialequations.Roundeachofyouranswerstothenearesthundredth.

(a) 39 250x− = (b) ( )50 2 1000x = (c) 105 35x

= 5. Solveeachofthefollowingexponentialequations.Becarefulwithyouruseofparentheses.Expresseach

answertothenearesthundredth.

(a) 2 56 300x− = (b) ( ) 131 1

2 6

x+= (c) ( )12500 1.02 2300

x

=


Ms.Talhami 17

APPLICATIONS

6. ThepopulationofRedHookisgrowingatarateof3.5%peryear.Ifitscurrentpopulationis12,500,inhowmanyyearswillthepopulationexceed20,000?Roundyouranswertothenearestyear.Onlyanalgebraicsolutionisacceptable.

7. Aradioactivesubstance isdecayingsuchthat2%of itsmass is losteveryyear. Originallytherewere50

kilogramsofthesubstancepresent.

REASONING8. Ifapopulationdoublesevery5years,howmanyyearswillittakeforthepopulationtoincreaseby10times

itsoriginalamount? First:Ifthepopulationgetsmultipliedby2every5years,whatdoesitgetmultipliedbyeachyear?Usethis

tohelpyouanswerthequestion.9. Findthesolutiontothegeneralexponentialequation ( )cxa b d= ,intermsoftheconstantsa,c,dandthe

logarithmofbaseb.Thinkaboutreversingtheorderofoperationsinordertosolveforx.

(a) Write an equation for the amount,A, of thesubstanceleftaftert-years.

(b) Findtheamountoftimethatittakesforonlyhalf of the initial amount to remain. Roundyouranswertothenearesttenthofayear.


Ms.Talhami 18

THE NUMBER e AND THE NATURAL LOGARITHM COMMONCOREALGEBRAII

Therearemanynumbersinmathematicsthataremoreimportantthanothersbecausetheyfindsomanyusesineithermathematicsorscience.Goodexamplesofimportantnumbersare0,1,i,andπ .Inthislessonyouwillbeintroducedtoanimportantnumbergiventhelettereforits“inventor”LeonhardEuler(1707-1783).ThisnumberplaysacrucialroleinCalculusandmoregenerallyinmodelingexponentialphenomena.Exercise#1:Whichofthegraphsbelowshows xy e= ?Explainyourchoice.Checkonyourcalculator.(1) (2) (3) (4)Explanation:Veryofteneisinvolvedinexponentialmodelingofbothincreasinganddecreasingquantities.Thecreationofthesemodelsisbeyondthescopeofthiscourse,butwecanstillworkwiththem.Exercise#2:Apopulationofllamasonatropicalislandcanbemodeledbytheequation 0.035500 tP e= ,wheretrepresentsthenumberofyearssincethellamaswerefirstintroducedtotheisland.

THENUMBERe

1.Like ,eisirrational. 2.e 3.UsedinExponentialModeling

y

x

y

x

y

x

y

x

(a) How many llamas were initially introduced at?Showthecalculationthat leadstoyour

answer.

(b) Algebraicallydeterminethenumberofyearsforthe population to reach 600. Round youranswertothenearesttenthofayear.


Ms.Talhami 19

Becauseoftheimportanceof xy e= ,itsinverse,knownasthenaturallogarithm,isalsoimportant.Thenaturallogarithm,likealllogarithms,givesanexponentasitsoutput.Infact,itgivesthepowerthatwemustraiseetoinordertogettheinput.Exercise#3:Withouttheuseofyourcalculator,determinethevaluesofeachofthefollowing.

(a) ( )ln e (b) ( )ln 1 (c) ( )5ln e (d) ln e Thenaturallogarithmfollowsthethreebasiclogarithmlawsthatalllogarithmsfollow.Thefollowingproblemsgiveadditionalpracticewiththeselaws.

Exercise#4:Whichofthefollowingisequivalentto3

2ln xe

⎛ ⎞⎜ ⎟⎝ ⎠

?

(1) ln 6x + (3)3ln 6x −

(2)3ln 2x − (4) ln 9x − Exercise#5:Ahotliquidiscoolinginaroomwhosetemperatureisconstant.Itstemperaturecanbemodeledusingtheexponentialfunctionshownbelow.Thetemperature,T,isindegreesFahrenheitandisafunctionofthenumberofminutes,m,ithasbeencooling.

( ) 0.03101 67mT m e−= +

THENATURALLOGARITHM

Theinverseof :

(a) Whatwastheinitialtemperatureofthewaterat.Dowithoutusingyourcalculator.

(b)How do you interpret the statement that?

(c) Using the natural logarithm, determinealgebraicallywhenthetemperatureoftheliquidwill reach . Show the steps in yoursolution. Round to the nearest tenth of aminute.

(d)On average, how many degrees are lost perminuteovertheinterval ?Roundtothenearesttenthofadegree.


Ms.Talhami 20

THE NUMBER e AND THE NATURAL LOGARITHM COMMONCOREALGEBRAIIHOMEWORK

FLUENCY

1. Whichofthefollowingisclosesttothey-interceptofthefunctionwhoseequationis 110 xy e += ?

(1)10 (3)27

(2)18 (4)522. Onthegridbelow,thesolidcurverepresents xy e= .Whichofthefollowingexponentialfunctionscould

describethedashedcurve?Explainyourchoice.

(1) ( )12

xy = (3) 2xy =

(2) xy e−= (4) 4xy =

3. Thelogarithmicexpression3ln ey

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

canberewrittenas

(1)3ln 2y − (3) ln 62y −

(2)1 6ln2y− (4) ln 3y −

4. Whichofthefollowingvaluesoftsolvestheequation 25 15te = ?

(1) ln1510

(3) 2ln3

(2) 12 ln 5

(4) ln 32

5. Atwhichofthefollowingvaluesofxdoes ( ) 22 32xf x e= − haveazero?

(1) 5ln2 (3) ln8

(2) ln 4 (4) 2ln5

y =

y

x


Ms.Talhami 21

6. Fortheequation ctae d= ,solveforthevariabletintermsofa,c,andd.Expressyouranswerintermsofthenaturallogarithm.

APPLICATIONS7. Flu is spreading exponentially at a school. The number of new flu patients can bemodeled using the

equation 0.1210 dF e= ,wheredrepresentsthenumberofdayssince10studentshadtheflu. (a) Howmanydayswill ittakeforthenumberofnewflupatientstoequal50?Determineyouranswer

algebraicallyusingthenaturallogarithm.Roundyouranswertothenearestday. (b) FindtheaveragerateofchangeofFoverthefirstthreeweeks,i.e.0 21d≤ ≤ .Showthecalculationthat

leadstoyouranswer.Giveproperunitsandroundyouranswertothenearesttenth.Whatisthephysicalinterpretationofyouranswer?

8. Thesavingsinabankaccountcanbemodeledusing .0451250 tS e= ,wheretisthenumberofyearsthemoney

hasbeenintheaccount.Determine,tothenearesttenthofayear,howlongitwilltakefortheamountofsavingstodoublefromtheinitialamountdepositedof$1250.


Ms.Talhami 22

COMPOUND INTEREST COMMONCOREALGEBRAII

Intheworldsofinvestmentanddebt,interestisaddedontoaprincipalinwhatisknownascompoundinterest.Thepercentrateistypicallygivenonayearlybasis,butcouldbeappliedmorethanonceayear.Thisisknownasthecompoundingfrequency.Let'stakea lookatatypicalproblemtounderstandhowthecompoundingfrequencychangeshowinterestisapplied.Exercise#1: A person invests $500 in an account thatearnsanominalyearlyinterestrateof4%.

So,thepatternisfairlystraightforward.Forashortercompoundingperiod,wegettoapplytheinterestmoreoften,butatalowerrate.Exercise#2:Howmuchwould$1000investedatanominal2%yearlyrate,compoundedmonthly,beworthin20years?Showthecalculationsthatleadtoyouranswer.(1)$1485.95 (3)$1033.87(2)$1491.33 (4)$1045.32Thispatternisformalizedinaclassicformulafromeconomicsthatwewilllookatinthenextexercise.Exercise#3:Foraninvestmentwiththefollowingparameters,writeaformulafortheamounttheinvestmentisworth,A,aftert-years. P=amountinitiallyinvested r=nominalyearlyrate n=numberofcompoundsperyear

(a) Howmuchwouldthisinvestmentbeworthin10years if thecompoundingfrequencywasonceperyear?Showthecalculationyouuse.

(b) If,ontheotherhand,the interestwasappliedfour times per year (known as quarterlycompounding),whywoulditnotmakesensetomultiplyby1.04eachquarter?

(c) Ifyouweretoldthataninvestmentearned4%per year, how much would you assume wasearnedperquarter?Why?

(d)Usingyouranswerfrompart(c),calculatehowmuchthe investmentwouldbeworthafter10years of quarterly compounding? Show yourcalculation.


Ms.Talhami 23

TherateinExercise#1wasreferredtoasnominal(innameonly).It'sknownasthis,becauseyoueffectivelyearnmorethanthisrateifthecompoundingperiodismorethanonceperyear.Becauseofthis,bankersrefertotheeffectiverate,ortherateyouwouldreceiveifcompoundedjustonceperyear.Let'sinvestigatethis.Exercise#4:Aninvestmentwithanominalrateof5%iscompoundedatdifferentfrequencies.Givetheeffectiveyearly rate,accurate to twodecimalplaces, foreachof the followingcompounding frequencies.Showyourcalculation.(a)Quarterly (b)Monthly (c)DailyWecouldcompoundatsmallerandsmallerfrequencyintervals,eventuallycompoundingallmomentsoftime.InourformulafromExercise#3,wewouldbelettingnapproachinfinity.Interestinglyenough,thisgivesrisetocontinuous compounding and theuseof thenatural basee in the famous continuous compound interestformula.Exercise #5: A person invests $350 in a bank account that promises a nominal rate of 2% continuouslycompounded.

CONTINUOUSCOMPOUNDINTEREST

Foraninitialprincipal,P,compoundedcontinuouslyatanominalyearlyrateofr,theinvestmentwouldbeworthanamountAgivenby:

(a) Write an equation for the amount thisinvestmentwouldbeworthaftert-years.

(b)Howmuchwouldtheinvestmentbeworthafter20years?

(c) Algebraicallydeterminethetimeitwilltakeforthe investment to reach $400. Round to thenearesttenthofayear.

(d)What is the effective annual rate for thisinvestment?Roundtothenearesthundredthofapercent.


Ms.Talhami 24

COMPOUND INTEREST COMMONCOREALGEBRAIIHOMEWORK

APPLICATIONS1. Thevalueofaninitialinvestmentof$400at3%nominalinterestcompoundedquarterlycanbemodeled

usingwhichofthefollowingequations,wheretisthenumberofyearssincetheinvestmentwasmade? (1) ( )4400 1.0075 tA= (3) ( )4400 1.03 tA= (2) ( )400 1.0075 tA= (4) ( )4400 1.0303 tA= 2. Whichof the following represents thevalueof an investmentwithaprincipalof$1500withanominal

interestrateof2.5%compoundedmonthlyafter5years? (1)$1,697.11 (3)$4,178.22 (2)$1,699.50 (4)$5,168.71

3. Francoinvests$4,500inanaccountthatearnsa3.8%nominalinterestratecompoundedcontinuously.Ifhewithdrawstheprofitfromtheinvestmentafter5years,howmuchhasheearnedonhisinvestment?

(1)$858.92 (3)$922.50 (2)$912.59 (4)$941.62

4. Aninvestmentthatreturnsanominal4.2%yearlyrate,butiscompoundedquarterly,hasaneffectiveyearlyrateclosestto

(1)4.21% (3)4.27% (2)4.24% (4)4.32%

5. Ifaninvestment'svaluecanbemodeledwith12.027325 1

12

t

A ⎛ ⎞= +⎜ ⎟⎝ ⎠thenwhichofthefollowingdescribes

theinvestment? (1)Theinvestmenthasanominalrateof27%compoundedevery12years. (2)Theinvestmenthasanominalrateof2.7%compoundedever12years. (3)Theinvestmenthasanominalrateof27%compounded12timesperyear.(4)Theinvestmenthasanominalrateof2.7%compounded12timesperyear.


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6. Aninvestmentof$500ismadeat2.8%nominalinterestcompoundedquarterly.REASONING

7. Theformula 1ntrA P

n⎛ ⎞= +⎜ ⎟⎝ ⎠

canberearrangedusingpropertiesofexponentsas 1tnrA P

n⎛ ⎞⎛ ⎞= +⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

.Explain

whattheterm 1nr

n⎛ ⎞+⎜ ⎟⎝ ⎠

helpstocalculate.

8. The formula rtA Pe= calculates the amount an investment earning a nominal rate of r compounded

continuouslyisworth.Showthattheamountoftimeittakesfortheinvestmenttodoubleinvalueisgiven

bytheexpression ln 2r

.

(a) WriteanequationthatmodelstheamountAthe investment is worth t-years after theprincipalhasbeeninvested.

(b)Howmuch is the investmentworth after 10years?

(c) Algebraicallydeterminethenumberofyearsitwill take for the investment to be reach aworth of $800. Round to the nearesthundredth.

(d)Whydoes itmakemoresensetoroundyouranswerin(c)tothenearestquarter?Statethefinalanswerroundedtothenearestquarter.


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NEWTON'S LAW OF COOLING COMMONCOREALGEBRAII

Thetemperatureofacoolingliquidinalargeroomwithasteadytemperatureisagreatexampleofatypeofatransformedexponentialfunction.Wewillexplorethistodaytoseehowasimpleexponentialfunctioncanbeusedtobuildamorecomplexone.

Exercise#1:Considerthedecreasingexponentialfunction ( ) 182

x

f x ⎛ ⎞= ⎜ ⎟⎝ ⎠.

O.k.So,nowlet'strytomodelaliquid'stemperaturethatiscoolinginalargeroom.Exercise#2:Assumealiquidstartsatatemperatureof 200 F andbeginstocoolinaroomthatisatasteadytemperatureof70 F.

(a) Useyourcalculatortosketchthegraphusingthewindowindicated.

(b) Clearly the value of y gets smaller as x getslarger. Does it ever reach zero?Whyorwhynot?

10

5

x

y

(a) Draw a rough sketch of what you believe theliquid'stemperaturefunctionlookslikeastimeincreases.

(b) Basedonyourgraphfrom(a)andyourworkinExercise#1,whywouldanequationoftheform

notmodelthiscoolingwell(assumingthat )? How could we modify thisequation tomake itmodel the situationmorerealistically?

Time

Temperature


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Exercise#3:Let'sstickwiththesamecoolingliquidthatwehadbefore,i.e.onethatstartsat 200 F andcoolsdowninaroomthatisheldataconstanttemperatureof 70 F.WewillnowmodelthiscoolingFahrenheittemperatureusingtheequation ( ) ( )tF t a b c= + ,wherea,b,andcareallparameters(constants)inthemodelandtistimeinminutes.

(g) Algebraicallydetermine,tothenearesttenthofaminute,whenthetemperaturereaches100 F .

(a) Which of these constants is equal to 70 andwhy?Thinkaboutthelastproblem.

(b)None of these constants is equal to 200, but. What constant does this let you

solvefor?Finditsvalue.

(c) Inorderto findthevalueofbwhatadditionalinformationwouldweneed?

(d)Determine the value of b if the temperature,after5minutes,is .Roundtothenearesthundredth.

(e)Whatisthetemperatureoftheliquidafterhalfanhour?Anhour?Twohours?

(f) Using your calculator, sketch the graph of theliquid'stemperature.Decideonanappropriatewindowandlabelitonyouraxes.

Time

Temperature


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NEWTON'S LAW OF COOLING COMMONCOREALGEBRAIIHOMEWORK

FLUENCY1. Forthefunction ( ) ( )xf x a b= ,where 0 1b< < ,whatvaluedoesthefunction'soutputapproachasxgets

verylarge?2. Forthefunction ( ) ( )xg x a b k= + ,where 0 1b< < ,whatvaluedoesthefunction'soutputapproachasx

getsverylarge?3. Given ( ) ( )tf t a b c= + ,whichofthefollowingrepresentsthey-interceptofthisfunction?Showhowyou

arrivedatyourchoice. (1) a (3) c (2) a c+ (4)b c+ APPLICATIONS4. Aliquidstartsataninitialtemperatureof175 Candcoolsdowninaroomheldataconstanttemperature

of16 C .It'stemperaturecanbemodeled,asafunctionoftimecooling,bytheequation ( )xy a b c= + .Whichofthefollowingstatementsistrue?

(1) 159 and 16a c= = (3) 175 and 16a c= = (2) 16 and 159a c= = (4) 16a = and 175c = 5. Acoolingliquidhasatemperaturegivenbythefunction ( ) ( )132 .83 40mT m = + ,wheremisthenumberof

minutesithasbeencooling.Whichofthefollowingtemperaturesdidtheliquidstartat? (1)40 (3)172 (2)92 (4)132


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6. A liquidstartsata temperatureof190 F andcoolsdown inaroomheldataconstant 65 F .After10minutesofcooling, it isatatemperatureof 92 F .TheFahrenheittemperature,F,canbemodeledasafunctionoftimeinminutes,t,bytheequation:

( ) ( )tF t a b c= +

(a) Determinethevaluesoftheparametersa,b,andc.Roundthevalueofbtothenearesthundredth.Statetheequationofyourfinalmodel.Showtheworkthatleadstoeachofyouranswers.

(b) Algebraically,determinethenumberofminutesitwilltakeforthetemperaturetoreach 70 F.Round

tothenearesttenthofaminute.REASONING7. Whenwemodelthetemperatureofacoolingliquidusingtheequation ( )xT a b c= + ,wehavelearnedthat

thevalueofcrepresentsthesteadytemperatureoftheroom.Thequantity ( )xa b doesmodelsomethingphysically.Canyoudeterminewhatitis?

8. AWarming Liquid - A liquid is taken out of a refrigerator and placed in a warmer room, where its

temperature,inF,increasesovertime.Itcanbemodeledusingtheequation ( ) ( )74 39 0.87 mT m = − .

(a) What temperature did the liquid start at?Showtheworkthatleadstoyouranswer.

(b)Whatisthetemperatureoftheroom?


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ASYMPTOTES COMMONCOREALGEBRAII

Sometypesoffunctionsbehavelikeeitherhorizontalorverticallinesinportionsoftheirdomains.Whenthisoccurs, we say that the function has an asymptote. In this course, we will only study the asymptotes ofexponentialandlogarithmicfunctions.Exercise#1:Giventheshiftedexponentialfunction ( ) ( )8 .75 3xf x = + answerthefollowingquestions.(a) Usingyourgraphingcalculator,drawasketchofthe

functionontheaxesshownattheright.(b) By examining this graph and a table in your

calculatorwhaty-valuedoesthisgraphnevergobelow?

(c) Explain why the function's output can never

evenreachthey-valueyouspecifiedin(b).Exercise#2:Giventheexponentialfunction ( )18 .98 4xy = +

(d)Givetheequationofthehorizontalasymptoteofthisfunctionanddrawitonyourcalculator.

HORIZONTALASYMPTOTESOFEXPONENTIALFUNCTIONS

Forthefunction ,thehorizontalline willbeahorizontalasymptoteofthefunction.

(a) What is they-interceptof this function?Showhowyouarrivedatyouranswer.

(b)Whatistheequationofthisfunction'shorizontalasymptote?


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Exponentialfunctionshavehorizontalasymptotes.Logarithmicfunctionsareawholedifferentmatter.

Exercise #3: Given the logarithmic function 2logy x= and its graph pictured below, answer the followingquestions.(a) Whatistheequationofthisfunction'sinverse?Whatis

theequationofitsinverse'shorizontalasymptote?(b)Whydoes itmakesensethatthe logarithmic function

shown would have a vertical asymptote instead of ahorizontalone?Whatisitsequation?

Exercise#4:Giventhefunction ( )3log 9y x= − ,answerthefollowingquestions.

(c) Whywouldtherelatedfunction ( )3log 9 5y x= − + havethesameverticalasymptote?Exercise#5:Foreachofthefunctionslistedbelow,statewhattypeofasymptoteithasandgiveitsequation.

(a) ( )5 1.075 13xy = + (b) ( )56log 2y x= + (c) ( )ln 7y x= +

VERTICALASYMPTOTESOFLOGARITHMICFUNCTIONS

Forthefunction ,theverticalline willbeaverticalasymptoteofthefunction.

(a) Whywon'tthisfunctionhaveay-intercept? (b)What is theequationofthis function'sverticalasymptote?Howcanyoudeterminethis?


Ms.Talhami 32

ASYMPTOTES COMMONCOREALGEBRAIHOMEWORK

FLUENCY1. Foreachofthefollowingexponentialfunctions,statetheequationofitshorizontalasymptote. (a) 3 7xy = + (b) ( )8 .5 2xy = − (c) .3510 5xy e= + (d) .0840 11.5ty e−= + (e) ( )120 .95 72xy = + (f) ( )14 5 1.06 ty = + 2. Foreachofthefollowinglogarithmicfunctions,statetheequationofitsverticalasymptote. (a) ( )logy x= (b) ( )3log 5y x= − (c) ( )5ln 2y x= + (d) ( )12log 3 7y x= − + (e) ( )4ln 5y x= + (f) ( )510log 4 8y x= + + 3. Forthefunction ( )2log 8 10y x= + − whichofthefollowingistrue? (1)Thefunctionhasay-interceptof 10− . (2)Thefunctionhasahorizontalasymptoteof 10y = − .

(3)Thefunctionhasaverticalasymptoteof 8x = − . (4)Thefunctionhasadomainofallrealnumbers.4. Ifthefunction ( )ln 10 3y x= − + isshiftedfourunitstotherightandtwounitsdownthenitwillhavean

asymptoteof (1) 14x = (3) 1y = (2) 6x = − (4) 5y = −


Ms.Talhami 33

5.Whichofthefollowingfunctionswouldhaveaverticalasymptoteof 4x = ? (1) 2 4xy = + (3) ( )ln 4y x= + (2) 4 2xy = + (4) ( )ln 4y x= − APPLICATIONS6. Aliquidiscoolingsuchthatitstemperature,indegreesFahrenheit,isgivenbythefunction:

( ) .12128 74mT m e−= +

wheremisthenumberofminutestheliquidhasbeencooling.

(c) Onaverage,howfastistheliquidcoolinginthefirsthalf-hour,indegreesperminute?Onaverage,how

fast is itcooling inthesecondhalf-hour, indegreesperminute?Roundbothanswerstothenearesthundredth.

(d) Thegraphofthefunctionisshownattherightwith

pointsat 0, 30, and 60t = minutes.Explainwhatishappeningin(c)totheaveragerateofcoolingbasedonthegraphanditsasymptote.

(a) What is the initial temperature of the liquidwhenitbeginstocool?

(b)What is the equation of this function'sasymptote?Whatdoesitrepresent?

Algebra 2 Unit 4B: Logarithms - Ms. Talhami · Algebra 2 Unit 4B: Logarithms Ms. Talhami 3 Exercise #2: Evaluate the following logarithms. If needed, write an equivalent exponential

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