Logarithms And Exponential Functions Study Guide

Logarithms and Exponential FunctionsStudy GuideExponential ModelsClues in the word problems tell you which formula to use. If there’s no mention of compounding, use agrowth or decay model. If your interest is compounded, check for the word continuous. That’s your clue touse the “Pert” Formula.Simple InterestGrowthSimple InterestDecay𝐴(𝑡) 𝑎(1 𝑟)𝑡𝐴(𝑡) 𝑎(1 𝑟)𝑡𝐴(𝑡) Amount after time t.Time𝑡𝑎𝑛Compound Interest𝑟 𝑛𝑡𝐴(𝑡) 𝑃 (1 )𝑛Initial amountNumber of interest payments inone yearGrowthExample baseball card bought for 150 increases invalue at a rate of 3% each year. How much is the cardworth in 10 years?𝐴 150(1 .03)10DecayYou bought a new Ford truck for 40,000 yesterday.The truck depreciates a rate of 11% each year. Howmuch is your truck worth 8 years from �) 𝑃𝑒 𝑟𝑡Rate expressed as a decimalInitial investment1.) The yellow bellied sapsucker has a populationgrowth rate of approximately 4.7% If the populationwas 8,530 in 2000 and this growth rate continues,about how many yellow bellied sapsuckers will therebe in 2006?2.) Amy Farah Fowler bought a new car for 25,000.Suppose the car depreciates at a rate of 13% per year. Howmuch will the car be worth in 4 years?𝐴 40000(1 .11)8Compound InterestYour favorite Aunt gives you a quick pick. It’s yourlucky day! You win 1500. You give 500 to your Auntand put the rest in a savings account that pays 3%interest compounded monthly. How much money willyou have in 10 years?𝐴 1000(1 3.) If you put 2400 in an account that pays 6.2% interestcompounded quarterly. How much will you have in eightyears?. 03 (12)(10))12Continuous CompoundingYour Aunt decides to deposit the 500 you gave her intoa savings account at her bank. This account pays 3.5%interest and compounds continuously. How muchmoney will she have in this account in 8 years?4.) If you put the same 2400 in an account that pays 5.7%interest compounded continuously. How much will youhave in eight years?𝐴 500𝑒 (.035)(8)1

Logarithms and Exponential FunctionsInverse FunctionsTo find the inverse of a function,1. Switch x and y values2. Solve for yStudy GuideFind the inverse of each function:𝑥5.) 𝑓(𝑥) 2𝑥 2 86.) 𝑓(𝑥) 4 3Inverse notation: 𝑓 1 (𝑥)For logs and exponents, put theequation in the “other form”. Thenswitch x and y, solve for y.y log4(16x)x log4(16y)4x 16y4x/16 y4x-2 yFind the inverseSwitch x and yPut in exponent formSolve for ySimplify if possibley 4xx 4yy log4xFind the inverseSwitch x and yPut in log formDefinition of LogarithmsTHE Relationship𝐼𝑓 𝑦 𝑏 𝑥 , 𝑡ℎ𝑒𝑛 𝑙𝑜𝑔𝑏 𝑦 𝑥7.) 𝑓(𝑥) 3𝑥 28.) 𝑓(𝑥) log (2𝑥 1)Write the following in log form:9.) 62 3610.) 53 12511.) 24 32Write 62 36 in log form𝑙𝑜𝑔6 36 2Write 𝑙𝑜𝑔2 64 6 in exponential formWrite the following in exponential form:12.) log 2 8 313.) log 3 81 426 6414. ) log 4 16 2Change of base formulalog 𝑥𝑙𝑜𝑔𝑏 𝑥 log 𝑏Evaluate 𝑙𝑜𝑔6 32𝑙𝑜𝑔6 32 Common log𝑙𝑜𝑔10 𝑖𝑠 𝑤𝑟𝑖𝑡𝑡𝑒𝑛 𝑎𝑠 𝑙𝑜𝑔15.) Evaluate 𝑙𝑜𝑔2 8Natural Log𝑙𝑜𝑔𝑒 𝑖𝑠 𝑤𝑟𝑖𝑡𝑡𝑒𝑛 𝑎𝑠 𝑙𝑛16.) Evaluate 𝑙𝑜𝑔0.25 0.0625log 32 1.9343log 62

Logarithms and Exponential FunctionsProperties of LogarithmsPROPERTIESEX 1: Condense the following into onelog statement.𝑙𝑜𝑔𝑏 𝑏 1𝑙𝑜𝑏𝑏 1 0𝑙𝑜𝑔𝑏 𝑚𝑛 𝑙𝑜𝑔𝑏 𝑚 𝑙𝑜𝑏𝑏 𝑛𝑙𝑜𝑔𝑏𝑚 𝑙𝑜𝑔𝑏 𝑚 𝑙𝑜𝑔𝑏 𝑛𝑛𝑙𝑜𝑔𝑏 𝑚𝑛 𝑛 𝑙𝑜𝑔𝑏 𝑚To condense log statements, theymust have the same base.Study Guide3 𝑙𝑜𝑔4 𝑥 2 𝑙𝑜𝑔4 𝑦Step 1: Move the constants in front ofthe log statements into the exponentposition. 𝑙𝑜𝑔4 𝑥 3 𝑙𝑜𝑔4 𝑦 2Step 2: Combine the arguments.Change subtraction to multiplicationand addition to multiplication. 𝑙𝑜𝑔4 𝑥 3 𝑦 2𝑥EX2: Expand the expression 𝑙𝑜𝑔 𝑦𝑧2Step 1: Deal with the divisionoperation first. Split the argumentinto two logs.log 𝑥 𝑙𝑜𝑔 𝑦𝑧 2Step 2: Split any statements withmultiplication into additionoperations. Be sure to distribute thenegative from the division.log 𝑥 (log 𝑦 log 𝑧 2 )log 𝑥 log 𝑦 log 𝑧 2Step 3: Move any exponents in frontof the log statement.log 𝑥 log 𝑦 2 log 𝑧Condense the following Log Statements17.) log 5 4 log 5 31218.) log 3x log 3x3319.) log 3 2 x 5 log 3 y20.) log 5 y 4(log 5 r 2 log 5 t )Expand the following Log Statements21.) log 6x3 y22.) log 2xyz23.) log2rst5w3

Logarithms and Exponential FunctionsStudy GuideSolve Exponential and Logarithmic EquationsTo solve an exponential equation, take Solve the equation 3𝑥 2 5 74.the log of both sides, and solve for theSubtract 5variable.3𝑥 2 69.To solve a logarithmic equation,rewrite the equation in exponentialform and solve for the variable.Other helpful properties:𝑙𝑜𝑔𝑏 𝑏 𝑥 𝑥𝑏𝑙𝑜𝑔𝑏 𝑥 𝑥log (3𝑥 2 ) log 69(𝑥 2) log 3 log 69log 69log 3𝑥 2 3.85𝑥 5.85𝑥 2 from bothsides.Take thelog of bothsidesSimplifythe leftsideEvaluatelogsSolve the equation 𝑙𝑜𝑔2 4𝑥 54𝑥 254𝑥 32𝑥 16Put inexponentialform.Simplify rightsideDivide bothsides by log 4.Solve for xSolve the following equations24.) 8n 1 325.) 103 y 526.) 4x 5 1227.) log (2x 5) 328.) log 4x 229). 2 log (2x 5) 4Sequences and Series (see last page for complete list of formulas)Determine if each sequence is arithmetic or geometric. Then find the 13st term in each sequence.30). 9, 14, 19, 24 31). -1, 6, -36, 216, Evaluate the following series.32.) 13, 15, , 2333.) 35𝑛 1(5𝑛 2)32.) A board is made up of 9 squares. A certain number of pennies is placed in each square following a geometricsequence. The first square has 1 penny, the second has 2 pennies, the third has 4 pennies, etc. When every square isfilled, how many pennies will be used in total?4

Logarithms and Exponential FunctionsStudy GuideSequencesARITHMETICGEOMETRICSequences happen when you add numbers. Thenumber added is called the common difference.𝑑 𝑎𝑛 𝑎𝑛 1Sequences happen when you multiply numbers. Thenumber multiplied is called the common ratio.𝑎𝑛𝑟 𝑎𝑛 1Explicit Formula of a basic geometric sequence𝑎𝑛 𝑎1 (𝑟 𝑛 1 )Where 𝑛 is the number of the term in the sequenceand 𝑟 is the common ratio.Recursive Formula of an geometric sequence𝑎𝑛 𝑟 (𝑎𝑛 1 )Where 𝑛 is the number of the term in the sequenceand 𝑑 is the common ratio.Explicit Formula of a basic arithmetic sequence𝑎𝑛 𝑎1 (𝑛 1)𝑑Where 𝑛 is the number of the term in the sequenceand 𝑑 is the common difference.Recursive Formula of an arithmetic sequence𝑎𝑛 𝑎𝑛 1 𝑑Where 𝑛 is the number of the term in the sequenceand 𝑑 is the common difference.SeriesA series is the sum of the terms in a sequence.Explicit Formula for the partial sum of an arithmeticExplicit Formula for the partial sum of a geometricsequencesequence1 𝑟𝑛𝑆𝑛 𝑎1 ()𝑛1 𝑟𝑆𝑛 (𝑎1 𝑎𝑛 )2𝑎1 𝑎𝑛 (𝑟 𝑛 )𝑆𝑛 1 𝑟To find the number of terms in a finite seriesTo find the number of terms in a finite series𝑛 𝑎𝑛 𝑎1 1𝑑𝑛 𝑎𝐿𝑜𝑔 ( 𝑎𝑛 )1𝐿𝑜𝑔(𝑟) 1Sigma NotationThe Greek letter sigma means to sum up. The example below is a simple summation.4 𝑛 1 2 3 4𝑛 1When a series is expressed in sigma notation, we translate it into the explicit formula to calculate the sum.𝑘𝑘𝑘 𝑎𝑛 𝑆𝑘 (𝑎1 𝑎𝑘 )2 𝑎𝑛 𝑆𝑘 𝑎1 (𝑛 1𝑛 11 𝑟𝑘)1 𝑟TI-84 Graphing Calculator[2nd][STAT] Math [5] [STAT] Ops [5] Expression [,] Variable[,] start[,] endOr for newer operating systems[MATH] [0]5

Logarithms and Exponential Functions Study Guide 2 Inverse Functions To find the inverse of a function, 1. Switch x and y values 2. Solve for y 6 Inverse no tation: 5( T) Find the inverse of each function: 5.) ( T) 2 T 8 6.) ( T) 8 3 For logs