NAME: ___________________ PERIOD: ___ DATE: ______________ PRE-CALCULUS

MR. MELLINA

UNIT 1: REVIEW OF ALGEBRA PART C: EXPONENTIAL & LOGARITHMIC FUNCTIONS

• Lesson1:Radicals&RationalExponents

• Lesson2:ExponentialFunctions• Lesson3:ApplicationsofExponentialFunctions

• Lesson4:Common&NaturalLogarithmicFunctions• Lesson5:PropertiesandLawsofLogarithms

• Lesson6:SolvingExponential&LogarithmicEquations• Lesson7:Exponential,Logarithmic,andOtherModels

Lesson1:PolynomialFunctions&OperationsObjectives:

• Defineandapplyrationalandirrationalexponents• Simplifyexpressionscontainingradicalsorrationalexponents.

WarmUp🔥 Simplify

a. b.!"#$%"

&!"#'$%(∙!$#(%'*

+&

Example1:OperationsonnthRootsSimplifyeachexpression.a. 8 ∙ 12 b. 12 − 75 c. 8𝑥3𝑦5( d. 5 + 𝑐 5 − 𝑐

Example2:SimplifyingExpressionswithRationalExponentsWritetheexpressionusingonlypositiveexponents

a. 8𝑟("𝑠+:

&/:

Example3:SimplifyingExpressionswithRationalExponentsSimplify

a. 𝑥*$ 𝑥

(" − 𝑥

($ b. 𝑥

<$𝑦5 𝑥𝑦

="+&

Example4:SimplifyingExpressionswithRationalExponentsLetkbeapositiverationalnumber.Writethegivenexpressionwithoutradicals,usingonlypositiveexponents.a. 𝑐?@*A 𝑐+@ B/&

Rational Exponents √𝑥!D =

Example5:RationalizingtheDenominatorRationalizethedenominatorofeachfraction.a. F

? b. &

:G 3

Example6:RationalizingtheNumeratorRationalizethenumeratorgivenℎ ≠ 0.

a.KGL+ K

L

Exponential Functions Have the form ___________ where and the base b is a positive number other than 1. b in the exponential function is called the _______________________. Exponential Growth: Exponential Decay: When b is greater than ______. When b is between ____ and ____. The x-axis is a horizontal __________________ of the graph this is an imaginary line that a graph approaches more and more closely. The Domain consists of the _____- values of the function. D: The Range consists of the _____ - values of the function. R:

Lesson2:ExponentialFunctionsObjectives:

• Graphandidentifytransformationsofexponentialfunctions.• Useexponentialfunctionstosolveapplicationproblems.

WarmUp🔥 Simplifya. 5+& b. 8&/: c. −3 ∙ 4:/&

Example1:GrowthorDecay?Withoutgraphing,determinewhethereachfunctionrepresentsexponentialgrowthorexponentialdecay.

a. b. c.

d. e. f.

Example2:GraphingGraphthefollowingexponentialgrowthfunctions.a. GrowthorDecay?Whattransformationshaveoccurredfromtheparentfunction?

e.

GrowthorDecay?Whattransformationshaveoccurredfromtheparentfunction?

Example3:PopulationGrowthIfthepopultionoftheUScontinuestogrowasithassince1980,thentheapproximatepopulation,inmillions,oftheUSinyeart,wheret=0correspondstotheyear1980,willbegivenbythefunction𝑃 𝑡 = 227𝑒R.RRT:U.a. Estimatethepopulationin2015.b. Whenwillthepopulationreachhalfabillion?

Example4:LogisticModelThepopulationoffishinacertainlakeattimetmonthsisgivenbythefunctionbelowwhere𝑡 ≥ 0.Thereisanupperlimitonthefishpopulationduetotheoxygensupply,availablefood,etc.Findtheupperlimitonthefishpopulation.

a. 𝑝 𝑡 = &R,RRRBG&5Y'Z/" b. 𝑝 𝑡 =&,RRR

BGBTTY'A.

Lesson3:ApplicationsofExponentialFunctionsObjectives:

• Createanduseexponentialmodelsforavarietyofexponentialgrowthanddecayapplicationproblems.

WarmUp🔥 Withoutagraph,determinewhetherthefunctioniseven,odd,orneithera. 𝑓 𝑥 = 10K b. 𝑓 𝑥 = Y

\GY'\

& c. 𝑓 𝑥 = Y

\+Y'\

&

Example1:FindthebalanceinanaccountusingCompoundinterest.Youdeposit$4000inanaccountthatpays2.92%annualinterest.Findthebalanceafter1yeariftheinterestiscompoundedwiththegivenfrequency.a. Quarterly b. Daily

CompoundInterestConsideranintialprincipalP(starting$)depositedinanaccountthatpaysinterestatanannualrater(expressedasadecimal),compoundedntimesperyear.TheamountAintheaccountaftertyearsisgivenbythisequation:

Example2:FindthebalanceinanaccountusingCompoundinterest.Youdeposit$7,000inanaccountthatpays.3%annualinterest.Findthebalanceafter1yeariftheinterestiscompoundedwiththegivenfrequency.a. Monthly b. Semi-Annuallyc. Yearly d. Weekly

Example3:ModelContinuouslyCompoundedInteresta. Youdeposit$4,000inanaccountthatpays6%annualinterestcompounded

continuously.Whatisthebalanceafter1year?

Continuously Compounded Interest When interest is compounded __________________, the amount A in an account after t years is given by the formula: Where P is the principal and r is the annual interest rate expressed as a decimal.

b. Youdeposit$2,500inanaccountthatpays5%annualinterestcompoundedcontinuously.Findthebalanceaftereachamountoftime.

v 2years

v 5years

v 7.5years

Example4:RadioactiveDecayWhenalivingorganismdies,itscarbon-14decaysexponentially.Anarcheologistdeterminesthattheskeletonofamastodonhaslost64%ofitscarbon-14.Thehalf-lifeofcarbon-14is5730years.a. Estimatehowlongagothemastodondied.

Half Life The amount of a radioactive substance that remains is given by the function

𝑓(𝑥) = 𝑃(0.5)K/L where P is the initial amount of the substance of the substance, x = 0 corresponds to the time when the radioactive decay began, and h is the half-life of the substance.

Lesson4:CommonandNaturalLogarithmicFunctionsObjectives:

• Evaluatecommonandnaturallogarithmswithandwithoutacalculator.• Solvecommonandnaturalexponentialandlogarithmicequationsbyusingan

equivalentequation.• Graphandidentifytransformationsofcommonandnaturallogarithmic

functions.

WarmUp🔥 Solveforx.a. 2K = 4 b. 10K = B

BR

Example1:EvaluatingCommonLogarithmsWithoutusingacalculator,findeachvalue.a. log 1000 b. log 1 c. log 10 d. log −3 Example2:UsingEquivelantStatementsSolveeachequationbyusinganequivelantstatement.a. log 𝑥 = 2 b. 10K = 29c. ln 𝑥 = 4 d. 𝑒K = 5

Example3:TransformingLogarithmicFunctionsDescribethetransformationsfromtheparentfunctiontothefunctionprovided.Sketchthegraphandgivethedomainandrangeofh.a. ℎ 𝑥 = 2 log 𝑥 − 3 b. ℎ 𝑥 = ln 2 − 𝑥 − 3 Example4:SolvingLogarithmicEquationsGraphicallyIfyouinvestmoneyataninterestrater,compoundedannually,thenD(r)givesthetimeinyearsthatitwouldtaketodouble.

𝐷 𝑟 =ln 2

ln(1 + 𝑟)

a. Howlongwillittaketodoubleaninvestmentof$2500at6.5%annualinterst?b. Whatannualinterestrateisneededinorderfortheinvestmentinpartatodoublein6

years?

Lesson5/5A:PropertiesandLawsofLogarithms/OtherBasesObjectives:

• Usepropertiesandlawsoflogarithmstosimplifyandevaluateexpressions.• Evaluatelogarithmstoanybasewithandwithoutacalculator.• Solveexponentialequationstoanybasebyusinganequivalentequation.• Usepropertiesandlawsoflogarithmstosimplifyandevaluatelogarithmic

expressionstoanybase.

WarmUp🔥 Evaluatethelogarithma. log 10 b. ln 𝑒& c. log 1000Example1:SolvingEquationsbyUsingPropertiesofLogarithmsUsethebasicpropertiesoflogarithmstosolve.a. ln 𝑥 + 1 = 2Example2:UsingtheProductLawofLogarithmsUsetheProductLawofLogarithmstoevaluateeachlogarithm.a. Givenlog 3 = 0.4771, log 11 = 1.0414.Findlog 33b. Givenln 7 = 1.9459, ln 9 = 2.1972.Findln 63

Example3:UsingtheQuotientLawofLogarithmsUsetheQuotientLawofLogarithmstoevaluateeachlogarithm.a. Givenlog 28 = 1.4472andlog 7 = 0.8451.Findlog 4b. Givenln 18 = 2.8904, ln 6 = 1.7918.Findln 3Example4:UsingthePowerLawofLogarithmsUsethePowerLawofLogarithmstoevaluateeachlogarithm.a. Givenlog 6 = 0.7782,findlog 6b. Givenln 50 = 3.9120,findlog 50( Example5:SimplifyingExpressionsWriteasasinglelogarithma. ln 3𝑥 + 4 ln 𝑥 − ln 3𝑥𝑦 b. log: 𝑥 + 2 + log: 𝑦 − log: 𝑥& − 4

c. 3 − log? 125𝑥 d. SimplifylnKK

+ ln 𝑒𝑥&" Example6:EvaluatingLogarithmstoOtherBasesWithoutusingacalculator,findeachvalue.a. log& 16 b. logB/: 9 c. log? −25 Example7:SolvingLogarithmicEquationsSolveeachequationforx.a. log? 𝑥 = 3 b. log3 1 = 𝑥c. logB/3 −3 = 𝑥 d. log3 6 = 𝑥

e. log: 𝑥 − 1 = 4Example8:ChangeofBaseFormulaEvaluatea. logf 9

Lesson6:SolvingExponentialandLogarithmicEquationsObjectives:

• Solveexponentialandlogarithmicequations• Solveavarietyofapplicationproblemsbyusingexponentialandlogarithmic

equations.

WarmUp🔥 Findtheinverse.a. 𝑓 𝑥 = log 𝑥 − 2 b. 𝑔 𝑥 = 𝑒&K+B Example1:PowersoftheSameBaseSolveforx.a. b. c. d.

e. f.

Example2:PowersofDifferentBases.Solveforxa. b. c. d. 25K+B = 3B+KExample3:UsingSubstitutionSolvea. 𝑒K − 𝑒+K = 4

Example4:UseanexponentialmodelYoudeposit$100inanaccountthatpays6%annualinterest.Howlongwillittakeforthebalancetoreach$1,000foreachgivenfrequencyofcompounding.a. Annual b. QuarterlyExample5:EquationswithLogarithms(Mixed)Solvea. b.

c. d. e. f.

g. h. Example6:EquationswithLogarithmsandConstantTermsSolvea. ln 𝑥 − 3 = 5 − ln 𝑥 − 3 b. log 𝑥 − 16 = 2 − log 𝑥 − 1

Lesson7:Exponential,Logarithmic,&OtherModelsObjectives:

• Modelrealdatasetswithpower,exponential,logarithmic,andlogisticfunctions.

Example1:FindingaModelIntheyearsbeforetheCivilWar,thepopulationoftheUSgrewrapidly,asshowninthefollowingtable.Findamodelforthisgrowth.a. Year Population

inmillions1790 3.931800 5.311810 7.241820 9.641830 12.861840 17.071850 23.191860 31.44

Example2:FindingaModelAftertheCivilWar,thepopulationoftheUScontinuedtoincrease,asshowninthefollowingtable.Findamodelforthisgrowth.a.Year Population

inmillions1870 38.561880 50.191890 62.981900 76.211910 92.231920 106.021930 123.201940 132.161950 151.131960 179.321970 202.301980 226.541990 248.72

Example3:FindingaModelThelengthoftimethataplanettakestomakeonecompleterotationaroundthesunisthaplanet’s“year”.Thetablebelowshowsthelengthofeachplanet’syear,relativetoanEarthyear,andtheaveragedistanceofthatplanetfromtheSuninmillionsofmiles.FindamodelforthisdatainwhichxisthelengthoftheyearandyisthedistancefromtheSun.a.Planet Year Distance Mercury 0.24 36.0Venus 0.62 67.2Earth 1.00 92.9Mars 1.88 141.6Jupiter 11.86 483.6Saturn 29.46 886.7Uranus 84.01 1783.0Neptune 164.79 2794.0Pluto 247.69 3674.5

Example4:FindingaModelFindamodelforpopulationgrowthinAnaheimCaliforniagiventheinformationinthefollowingtable.a.Year 1950 1970 1980 1990 1994Population 14,556 166,408 219,494 266,406 282,133