Unit 6 Exponential And Logarithmic Functions

Unit 6 Exponential and Logarithmic FunctionsLesson 1: Graphing Exponential Growth/Decay FunctionLesson Goals: Identify transformations of exponential functions Identify the domain/range and key features of exponential functionsWhy do I need to Learn This? Many real life applications involve exponential functions. Visually the graph can help you understand a problem better.Lesson 2: Exponential ApplicationsLesson Goals: Set up an exponential model for a real-life situation Understand the difference between a linear growth/decay and exponential growth/decay Solve financial equations involving simple and compound interestWhy do I need to Learn This? There are many real-life examples modeled by exponential growth or decay. Many loans and bank interest formulas use exponential growth.Lesson 3: Defining and Evaluating Logarithmic FunctionsLesson Goals: Rewrite between exponential and logarithmic forms of an expression Evaluate simple logarithmic expressions without a calculator Use a formula modeling a real-life situation that incorporates a logarithm.Why do I need to Learn This? Logarithms are used in a variety of scientific applications.Lesson 4: Properties of LogarithmsLesson Goals: Expand or condense logarithmic expressions in order to evaluate or simplify. Use the change-of-base formula to find decimal approximations of logarithms. Use formulas modeling real-life situation that incorporates a logarithm.Why do I need to Learn This? Logarithms are used in a variety of scientific applications.Lesson 5: Solving Exponential and Logarithmic EquationsLesson Goals: Solve an exponential equation by rewriting in log form or using inverse operations. Solve a log equation by rewriting in exponential form or using inverse operations. Solve a real-world problem modeled by an exponential or logarithmic equation.Why do I need to Learn This? Exponential Function and logarithms are used in a variety of scientific and financialapplications.

Unit6Lesson1GraphingExponentialFunctionsv ExponentialFunction:afunctionintheform𝑓 𝑥 𝑏 ! sadifferentfunction𝒇 𝒙 𝟐𝒙𝒇 𝒙 𝟏 𝒙𝟐Domain:Domain:Range:Range:y- ‐intercept:y- ‐intercept:Endbehavioras𝑥 :Endbehavioras𝑥 :Endbehavioras𝑥 :Endbehavioras𝑥 :Base𝒆:Thefunction𝑓 𝑥 𝑒 !isanexponentialgrowthfunction.𝒆

(0,1)Step4:Graph2 referencepoints. Find(0,1)referencepointFROMTHEASYMPTOTE urredifcountingfromtheasymptote) e) ReflectthepointifnecessaryOR .Youcanalwaysmakeanx- ‐ytable!!!A.𝑔 𝑥 3 2!!! 1B.𝑔 𝑥 0.5 3! 1

II.ExponentialGrowthFunctionswithbase𝒆

III.ExponentialDecayFunctionsA.𝑔 𝑥 3B.𝑔 𝑥 2! !!!! 2! !!!! 4

ecayfunctionsAnexponentialfunctionhastheform𝑓 𝑡 𝑎(1 𝑟)!where𝑎 purchasedarareguitarin2012for 2:Thevalueofatruckpurchasednewfor ange.TheDowJonesindexfortheperiod1980- ‐2000canbemodeledby𝐷 𝑡 878𝑒 !.!"!! DowJonesindexfortheyear1993.

II.CompoundInterestSimple Interest: Interest earned on the principle amount of money.Compound Interest: Interest earned on the principle amount of money ANDon the accumulated interest of previous periodsPrinciple: The initial amount of money investedCommon words for Daily365𝒓𝑨 𝑷 𝟏 𝒏𝒏𝒕The more often interest iscompounded (reinvested),the more money you earn!A final amountP initial principle amountr interest rate (as a decimal)t time (years)n number of times per year interest is compoundedNotice that if the interest is compounded once annually, the following equation will be used:𝑨 𝑷(𝟏 𝒓)𝒕Example 4:A person invests 3500 in an account that earns 3% annual interest. How much money will the personhave in the account after 10 years?Example 5:A person invests 8000 in an account that earns 6.5% annual interest compounded daily. How muchmoney will the person have in the account after 3 years?

III.InterestCompoundedContinuously! !Recall that base e comes from 1 !𝐴 𝑃 1 ! !"!as 𝑥 and 𝑒 2.72.can be rewritten as 𝐴 𝑃 𝟏 𝟏 𝒓𝒏!𝒏    𝑃𝟏 𝒓!𝟏 𝒏𝒏As the value of n increases, the bolded part of this formula becomes e. This can berewritten as 𝐴 𝑃𝒆𝒓! .Example 6:A person invests 5000 in an account that earns 3.5% annual interest compounded continuously. Howmuch money is in the person’s account after 4 years?Example 7:A person invests 1550 in an account that earns 4% annual interest compounded continuously. Howmuch money is in the person’s account after 9 years?

eachother.Q:Whydoweneedlogarithms?A:Tofindnon- !2 4log ! 4 22! 8log ! 8 32? 6?between2and3log ! 6  ? uationinitsalternateform.

𝑏 1, 𝑏 0Special Logarithmic Values to MemorizeLogarithm of 1log ! 1 0 because 𝑏 ! 1example: log ! 1 0Logarithm of base blog ! 𝑏 1 because 𝑏! 𝑏example: log ! 4 1Inverse Functionslog ! 𝑏 ! 𝑥 and 𝑏 !"#! ! 𝑥because exponential and logarithmic functions are inversesexample: 6!"#! ! 8Evaluatetheexpressionwithoutacalculator.1.log ! 812.log ! 0.043.log ! 34.log ! 8!5.log ! 6!6.7!"#! !"7.If𝑓 𝑥   log!" 𝑥,  find𝑓 1000 , 𝑓 0.01 ,  and𝑓 10 .

Common LogarithmNatural Logarithmlog!" 𝑥 log 𝑥log ! 𝑥       ln 𝑥* If a base is NOT shown, we assume base 10.***If you see ln (natural log), we assume base e.**Evaluateusingyourcalculator.1.log 52.ln 0.1

stoexpandalogarithmicexpression.(a)log !!! !!!Assumexandyarepositive.(b)ln 𝑥 ! 𝑦 !

fpossible.(a)log 6 2 log 2 log 3(b)3 ln 3 ln 𝑥 ln 𝑥 ln 9(c)log ! 27 log ! 81(d)log !!!"   log ! 625

III.Usingthechange- ‐of- ‐baseformulatoevaluatelogarithms.Change-of-Base FormulaLet u, b, and c be positive numbers with 𝑏 1 and 𝑐 1.log ! 𝑢 To use your calculator, let(a)Evaluatelog ! 7.b 10 orlog ! 𝑢log ! 𝑐b e.!(b)Evaluatelog ! !".

OGARITHMS!!!1.4 8!!!Noticethat2! 4  and  2! 82.2! 73.10!!!! 4 214.10 5𝑒 !!5.6!!!! 10 3

nsthatdon’twork)1.log ! 5x 1 log ! x 7UsethefactthatBASESarethesame.2.log ! 3𝑥 1 23.2 log 5𝑥 log 𝑥 1

III.SolvingaReal- 212 F.Theroomtemperatureis70 F.Thecoolingrateofthestewis𝑟 vingtemperatureof100 F?Newton’sLawofCooling:𝑇 𝑇! 𝑇! 𝑒 !!" 𝑇!T theendingtemperature𝑇! theinitialtemperature(beginningtemperature)𝑇! theroomtemperaturer theconstantcoolingratet deledbytheequation:𝑀 0.291 ln 𝐸 earthquakerelease?3.Supposethat uarterly.The! !equation𝐴 𝑃 1 st 500.

Unit 6 Exponential and Logarithmic Functions Lesson 1: Graphing Exponential Growth/Decay Function Lesson Goals: Identify transformations of exponential functions Identify the domain/range and key features of exponential functions Why do I need to Learn This? Many real life applications involve exponential functions.