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

key

Lesson1:PolynomialFunctions&OperationsObjectives:

• Defineandapplyrationalandirrationalexponents• Simplifyexpressionscontainingradicalsorrationalexponents.

WarmUp! Simplify

a. b.!"#$%"

&!"#'$%(∙!$#(%'*+&

Example1:OperationsonnthRootsSimplifyeachexpression.a. 8 ∙ 12 b. 12 − 75 c. 82345( d. 5 + 7 5 − 7

2 2

3y27m4nzp4 z

2m6np2nm 2 n mp22m42m2 n p4

4m4Fpu

2oz 253 456 253 553

353

2 2yTy 25 c

Example2:SimplifyingExpressionswithRationalExponentsWritetheexpressionusingonlypositiveexponents

a. 88("9+:

&/:

Example3:SimplifyingExpressionswithRationalExponentsSimplify

a. 2*$ 2

(" − 2

($ b. 2

<$45 24

="+&

Example4:SimplifyingExpressionswithRationalExponentsLetkbeapositiverationalnumber.Writethegivenexpressionwithoutradicals,usingonlypositiveexponents.a. 7?@*A −7 B/&

Rational Exponents √2!D =

innX

8Ertsz

4 rtS

It Itx x xEy y

Z

2 Etc2X x y4ttx4 2 XE y Zyk

Eat

45kt of E

E E 2

cC Cc c

Example5:RationalizingtheDenominatorRationalizethedenominatorofeachfraction.a. F

? b. &:G 3

Example6:RationalizingtheNumeratorRationalizethenumeratorgivenℎ ≠ 0.a.

KGL+ KL

Is 3 Cvs I7 6221 6321

Txt Txrutx

x thIturi h

1Ittf

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:/&

2 Is F 2

2 2 4 z 133 2 3 318

32

y abgrowthdecay

I 0 I

asymptote

CN N

L Coca

Example1:GrowthorDecay?Withoutgraphing,determinewhethereachfunctionrepresentsexponentialgrowthorexponentialdecay.

a. b. c.

d. e. f.

Example2:GraphingGraphthefollowingexponentialgrowthfunctions.a. GrowthorDecay?Whattransformationshaveoccurredfromtheparentfunction?

e.

GrowthorDecay?Whattransformationshaveoccurredfromtheparentfunction?

growth growth decay

growth decay decay

neutron

J y 4.2 1 3stretchedverticallyby a factor of 4shifted right 7 down 3

YffaEeIefuxtl

y 3Ed 2 hstretchedverticallyby a factor of 4shifted left 1 down z

I

Example3:PopulationGrowthIfthepopultionoftheUScontinuestogrowasithassince1980,thentheapproximatepopulation,inmillions,oftheUSinyeart,wheret=0correspondstotheyear1980,willbegivenbythefunctionO P = 227QR.RRT:U.a. Estimatethepopulationin2015.b. Whenwillthepopulationreachhalfabillion?

Example4:LogisticModelThepopulationoffishinacertainlakeattimetmonthsisgivenbythefunctionbelowwhereP

≥

0.Thereisanupperlimitonthefishpopulationduetotheoxygensupply,availablefood,etc.Findtheupperlimitonthefishpopulation.

a. W P = &R,RRRBG&5Y'Z/" b. W P = &,RRR

BGBTTY'A.<<""Z

The Natural Base e The history of mathematics is marked by the discovery of special numbers such as . Another special number is denoted by the letter e. The number is called the natural base e or the Euler number after its discoverer, Leonhard Euler. The natural base e is ____________. It is defined as follows:

Other Exponential Functions In most real-world applications, populations cannot grow infinitely large. A logistic model, which is designed to model situations that have limited future growth due to a fixed area, food supply, or other factors.

irrational

2,015 1980 35 P355 314.3 millionppl

500 zz eo0093T

f 84.9 1980 84.9 2064.9 It will take until 2065

Z

44as E n e o as c no e o5544T o

so f zo ooo fish so Yfg 2ooofish

Lesson3:ApplicationsofExponentialFunctionsObjectives:

• Createanduseexponentialmodelsforavarietyofexponentialgrowthanddecayapplicationproblems.

WarmUp! Withoutagraph,determinewhetherthefunctioniseven,odd,orneithera. [ 2 = 10K b. [ 2 = Y\GY'\

& c. [ 2 = Y\+Y'\&

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

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

even if fC xodd Fox fC x

fCi f Ftl fcc Ifc c fcc 7 FC Lfa Fct e e e e e eye

z 2neither the e't I

even odd

A P Itn nt

365A 4ooo It 0292 A 4ooo It 0292

cis 365

pf 4,118.09 pf 4118.52

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.

A 7000 I t pf ooo I2

z

7212.9 I 7211 58

ice 52cmA 7000 I 1 of A 7000 I t

7213 127210

continuously

A Pert

A 4000e05 l 4205 08

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

v 2years

v 5years

v 7.5years

Example4:RadioactiveDecayWhenalivingorganismdies,itscarbon-14decaysexponentially.Anarcheologistdeterminesthattheskeletonofamastadonhaslost64%ofitscarbon-14.Thehalf-lifeofcarbon-14is5730years.a. Estimatehowlongagothemastondied.

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

[(2) = O(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.

A 2500 e Ost 2762 93

A 2500 e 05155 3270.06

A 2500 e oses 3637.49

f x P o.gs o

0.36P P o s 30

0.36 O 5 30 solve in calc

8445.6 yrs ago

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 2 = 2 b. 10K = 29c. ln 2 = 4 d. QK = 5

Z I

xcooo 10 213 o

o JTo 10 2 342 no solution

102 _x loglox log29Coo X log29

lne lu5e4Ins

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

d 8 = ln 2ln(1 + 8)

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

years?

In Lx 2 3

fauxzogc Inez 3

parent fu logx parent fu Ink2logLx 3 is stretchedvertically InLz x 3 is reflectedhorizontallyby 2 and translated right3 translatedright2 and down3

D 0.065 1 11.0067In 110.065

about 11yrs

6 II Agsolve in calcInc Ctr

re 0.1225

12.25 interest rate

Lesson5/5A:PropertiesandLawsofLogarithms/OtherBasesObjectives:

• Usepropertiesandlawsoflogarithmstosimplifyandevaluateexpressions.• Evaluatelogarithmstoanybasewithandwithoutacalculator.• Solveexponentialequationstoanybasebyusinganequivalentequation.• Usepropertiesandlawsoflogarithmstosimplifyandevaluatelogarithmic

expressionstoanybase.

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

z

logow3 3

e 2 xx Ie 2 I

log33 log Il 3 log11 1 log3I 0414 to 4771

I 5185

In63 In 7 9 In t In91 9459 2.1972

4 1431

Example3:UsingtheQuotientLawofLogarithmsUsetheQuotientLawofLogarithmstoevaluateeachlogarithm.a. Givenlog 28 = 1.4472andlog 7 = 0.8451.Findlog 4b. Given18 = 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 32 + 4 ln 2 − ln 324 b. log: 2 + 2 + log: 4 − log: 2& − 4

loge4 log Zf log28 log71 4472 O 845 I

o 6021lui82

In 3 In Ia InCc8 InLce

2 8904 l 79181 0986

logof log 2 log6 Co27820 3891

Info In50 3 13 In 50 Ig 3.9120I 304

In3xtlux4 1h3Xy log Cycxtz log31 2 4In 3 4 4 lnC3xyth 3524

93

in zu1093

109 2

c. 3 − log? 1252 d. Simplifyln KK + ln Q2&"

Example6:EvaluatingLogarithmstoOtherBasesWithoutusingacalculator,findeachvalue.a. log& 16 b. logB/: 9 c. log? −25 Example7:SolvingLogarithmicEquationsSolveeachequationforx.a. log? 2 = 3 b. log3 1 = 2c. logB/3 −3 = 2 d. log3 6 = 2

y3 logs125T logsx 1n I 2 the e5thx t In ex23 3 logs x lux t g Ine 2Inx

logsX Init 14 it Zinx

Inxt I t zinxI4

x x

z l6 1 9 5 253x 4 135 1

32

no solution

z

53 6 I125 X o

f 3 6 61

nosolution

e. log: 2 − 1 = 4Example8:ChangeofBaseFormulaEvaluatea. logf 9

34 X I34 1811 1

82

togalog8

l 0566

Lesson6:SolvingExponentialandLogarithmicEquationsObjectives:

• Solveexponentialandlogarithmicequations• Solveavarietyofapplicationproblemsbyusingexponentialandlogarithmic

equations.

WarmUp! Findtheinverse.a. [ 2 = log 2 − 2 b. g 2 = Q&K+B Example1:PowersoftheSameBaseSolveforx.a. b. c. d.

e. f.

X logCy 2 x e2y I

10 y z lnx 1ne2y i1nx 2y I

y 10 2 In xtc Zgy Inxti

2

6 4 433

344 331 1 402 ios2

1014 2 109 6

z4x 33 314 2 9 6

4 3 3 5x 83 x I5

225 213 34g 35 6

zzx 23 iz ya 5 6

62x xt33 3

i

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

In4 Incl Xing 1h35

cu4 lull lYn 1.61811 4 1.7300or4

5 1n Il 1h33 4 1 In2 Cl x 1h3

In 33 4x1nz cuz In3 Xln30.291651m11 4 1uz tx in 3 1h31 uz

x 4hr21 In3 1h3 Luz

ln3 tl 0.462841h2 In3

ex ex e e 4

e eo 4 exe 2 IfsIne In ZITS

e 2 1 4 e

EI IEE 3ex 4 TIKI Ai In12 Vs is an extraneous

2 Ci Solution 2 Vs Lo

ex 41252

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

c. d. e. f.

4T1000 100 it Jt 1000 100 it 041

o C o6 t lo 1 or5 t

logto t log 1 06 log10 4t logCleo's

I t log 1.06I 4Tlog 1.015

ICE 1 410g 1015log 1obj

39 517 yearsC 38.664yrs

e3 5 14 7 its

e3 ti 3 x izs4

109336 10932 log 43 5 14311

log18 loggia 5

c8

logged log 27 e2 3 2

e 2 3 x3

e3

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

1ogzx2 2x 3 In4 2 Inna s

23 2 2x In 441 5

0 2 2 8 In x _5o ex 4 cx 2 es

4 x 4 2extraneous

2 In x 3 5

In ex 3 I2

512e x 3

512 3e

tog x 16 1 cogCx c 2

log X 17 16 2

102 2 i x 1 6

o x Z c x 84O X zc x 4

x _21 4 4

Lesson7:Exponential,Logarithmic,&OtherModelsObjectives:

• Modelrealdatasetswithpower,exponential,logarithmic,andlogisticfunctions.

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

inmillions1790 3.931800 5.311810 7.241820 9.64

X ofyears after 1790

Exponential model1830 12.86 Y 3.9572 1.02991840 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 226.54

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 164.79 2794.0Neptune 164.79 2794.0Pluto 247.69 3674.5

ofyears after 1870logisticexp 2 9812 r 99695

gquartic r 9984 yIYI.fi oe.o4

o.a.gyygqqqY 7.02 157 4 1 2.08 154 3

t C0.0135 2 t i 55 37.213

poor regr

tool 357cg

r2 999999

Y 92.89326669

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

logarithmicmodelZ 9993

y 1643983.42ft 424768.97333 In X