Unit 7 Exponential Functions. Mrs. Valen+ne Math III
|
|
- Scott Bruce Garrison
- 6 years ago
- Views:
Transcription
1 Unit 7 Exponential Functions Mrs. Valen+ne Math III
2 7.1 Exponential Functions Graphing an Exponen.al Func.on Exponen+al Func+on: a func+on in the form y = ab x, where a 0, b > 0 and b 1 Domain is all real numbers Range of a parent func+on is y > 0 To graph an exponen+al func+on, make a table of points, plot them, and connect in a smooth curve. Example: What is the graph of y = 2 x?
3 7.1 Exponential Functions Iden.fying Exponen.al Growth and Decay Exponen+al Growth Value of x increases as y increases b is growth factor (b > 1) Exponen+al Decay value of x increases as y decreases b is decay factor (0 < b < 1) In both cases, y-intercept is (0,a), the domain is all real numbers, the range is y > 0, and the asymptote is y = 0. Examples: Which are growth and which are decay? y = 12(0.95) x decay y = 0.25(2) x growth You put $1000 into a college savings account for four years. The account pays 5% interest annually. à growth
4 7.1 Exponential Functions Modeling Exponen.al Growth For exponen+al growth and decay, b = 1 + r. If r > 0, it is the rate of increase or growth rate. If r < 0, it is the rate of decay. Exponen+al growth and decay can be modeled by: Example: You put $1000 into a college savings account for four years. The account pays 5% interest annual interest. How much money will be in the account ayer six years? a = 1000 r = 0.05 t = 6 A(6) = 1000 ( ) 6 A(6) = 1000 (1.05) 6 A(6) = $
5 7.1 Exponential Functions Using Exponen.al Growth Suppose you invest $1000 in a savings account that pays 5% annual interest. If you make no addi+onal deposits or withdrawals, how many years will it take for the account to grow to at least $1500? a = 1000 r = 0.05 A(t) ( ) t The account will not contain $1500 un+l the ninth year. AYer nine years, the balance will be $ Suppose you invest $500 in a savings account that pays 3.5% annual interest. When will the account contain at least $650?
6 7.1 Exponential Functions Wri.ng an Exponen.al Func.on Exponen+al func+ons are oyen discrete. To model a discrete solu+on using an exponen+al func+on in the form y = ab x, find the growth or decay factor, b. If you know y values for two consecu+ve x-values, you can find the rate of change, and therefore b. b = 1 + r The equa+on for the graph provided: = b = b = a = 1000 (y-intercept) y = 1000(1.045) x
7 7.1 Exponential Functions Example: The table shows the world popula+on of the Iberian lynx in 2003 and If this trend con+nues and the popula+on is decreasing exponen+ally, how many Iberian lynx will there be in 2014? b = b = 0.8 = = a(0.8) 0 (x = 0 for ini+al value) 150 = a y = 150(0.8) x x = = 11 y = 150(0.8) 11 y 13 Iberian lynx ley in 2014
8 7.2 Properties of Exponential Functions Graphing y = ab x. Recall that the factor a in y = ab x can stretch or compress, and possibly reflect the graph of the parent func+on y = b x. Example: graph y = 2 x and y = 3*2 x. The red graph is the parent func+on while the blue graph is the stretched func+on. Since 3 is posi+ve, the graph is not reflected on the x-axis.
9 7.2 Properties of Exponential Functions Transla.ng the Parent Func.on y = b x. Horizontal y = ab (x-h) shiys the func+on ley or right h spaces Ver+cal y = ab x + k shiys the func+on up or down k spaces Examples: Compare the graph of each func+on to its parent func+on. y = 2 (x 4) y = 20 (½) x + 10
10 7.2 Properties of Exponential Functions Using an Exponen.al Model All transforma+ons combined: y = ab (x h) + k Example: The best temperature to brew coffee is between 195 F and 205 F. Coffee is cool enough to drink at 185 F. The table shows temperature readings from a sample cup of coffee. How long doe it take for a cup of coffee to be cool enough to drink? Use an exponen+al model. Room temp: 68 F Plot the points in the calculator using STAT. In STAT, L3 = L2-68 Use ExpReg L1, L3 to find the exponen+al model. Translate the model ver+cally by 68 y = (0.956) x + 68 It will take about 3.1 min for the coffee to cool to 185 F
11 7.2 Properties of Exponential Functions Rewri.ng an Exponen.al Func.on The func+on P = 80(1.25) d models the popula+on of a city, in thousands, ayer d decades. What exponen+al func+on models the popula+on ayer t years? What is the annual growth rate of the city s popula+on? 10d = t, or d = 0.1t P = 80(1.25) 0.1t P = 80( ) t P = 80(1.023) t A(t) = a(1 + r) t b = 1 + r = 1 + r = r 2.3% is the annual growth rate
12 7.2 Properties of Exponential Functions Evalua.ng e x. Some+mes, a func+on can have an irra+onal base. The graph of has an asymptote of y = e. Recall that e Natural base exponen+al func+ons are exponen+al func+ons with base e. Used to describe con+nuous growth or decay. Otherwise have the same proper+es as other exponen+al func+ons. Examples: Evaluate the following: e 3, e 6, and e e. e e e e
13 7.2 Properties of Exponential Functions Con.nuously Compounded Interest The formula for con+nuously compounded interest: Example: Suppose you won a contest at the start of 5 th grade that deposited $3000 in an account that pays 5% annual interest compounded con+nuously. How much will you have in the account when you enter high school 4 years later? Express the answer to the nearest dollar. A = Pe rt A = 3000e (0.05)(4) A = 3000e (0.2) A $3664
14 7.3 Logarithmic Functions as Inverses Wri.ng Exponen.al Equa.ons in Logarithmic Form Logarithm base b of posi+ve number x: For b > 0, b 1, log b x = y if and only if b y = x Domain: x > 0 This is the inverse of exponen+al func+ons. Example: what is the logarithmic form of each equa+on? 100 = =3 4 x = b y then log b x = y log = 2 x = b y then log b x = y log 3 81 = 4
15 7.3 Logarithmic Functions as Inverses Evalua.ng a Logarithm What is the value of log 8 32? Write a logarithmic equa+on Convert to exponen+al form Re-write using like bases Use Power Property of Exponents Set the exponents equal to each other (bases are same) Solve for x log 8 32 = x 32 = 8 x 2 5 = (2 3 ) x 2 5 = 2 3x 5 = 3x 5/3 = x
16 7.3 Logarithmic Functions as Inverses Using a Logarithmic Scale Common logarithm is log 10 (can be wriuen without base) Some+mes, measures of physical phenomena have such wide range values that the values reported are logarithms of the values. When you use the logarithm of a quan+ty instead of the quan+ty, you are using a logarithmic scale (Ex Richter scale).
17 7.3 Logarithmic Functions as Inverses Example: In December 2004, an earthquake with magnitude 9.3 on the Richter scale hit off the northwest coast of Sumatra. In March 2005, another one hit Sumatra with magnitude 8.7. The formula below compares the intensity levels of earthquakes where I is the intensity level determined by a seismograph, and M is the magnitude on a Richter scale. How many +mes more intense was the December earthquake than the March earthquake?
18 7.3 Logarithmic Functions as Inverses Graphing a Logarithmic Func.on Logarithmic func+on is the inverse of an exponen+al func+on. Recall that the graphs of inverse func+ons are reflec+ons of each other across y=x. Example: What is the graph of y = log 3 x? Describe the domain and range and iden+fy the y-intercept and asymptote. Domain is x > 0. The range is all real numbers. No y-intercept. Asymptote is x = 0.
19 7.3 Logarithmic Functions as Inverses Transla.ng y = log b x Transformed version: y = a log b (x h) + k a stretches ( a >1) or shrinks ( a <1), and reflects (-) over x- axis. h translates horizontally. k translates ver+cally Example: How does the graph of y = log 4 (x 3) + 4 compare to the graph of the parent func+on?
20 7.4 Properties of Logarithms Simplifying Logarithms Proper+es of logarithms are derived from proper+es of exponents. Examples: What is each expression wriuen as a single logarithm? log 4 32 log 4 2 6log 2 x + 5 log 2 y = log = 2 = log 4 16 = log 2 x 6 + log 2 y 5 = log 2 x 6 y 5
21 7.4 Properties of Logarithms Expanding Logarithms These proper+es can be used to expand a single logarithm. Examples: What is each logarithm expanded? = log 4x log y = log 4 + log x log y = log 9 x 4 log = 4log 9 x log = log log 3 (3x 3) 2 = log log 3 (3x 3) = log log log 3 (x 1) = log 8 8 (3a 5 ) ½ = log log 8 (3 ½ a 5/2 ) = log log 8 3 ½ + log 8 a 5/2 = 1+ ½log / 2 log 8 a
22 7.4 Properties of Logarithms Using the Change of Base Formula Since the calculator only uses the common logarithm, you will need to be able to change bases to evaluate some of the logarithms. Examples: What is the value of each expression? log log 5 36 =
23 7.4 Properties of Logarithms Using a Logarithmic Scale The ph of a substance equals log [H + ], where [H + ] is the concentra+on of hydrogen ions. [H + a ] for household ammonia is [H + v ] for vinegar is 6.3 x What is the difference of the ph levels of ammonia and vinegar? ph = log [H + ] log [H + a ] ( log [H+ v ]) = log [H + a ] + log [H+ v ] = log [H + v ] log [H+ a ] = log (6.3x10 3 ) log = log log 10 3 log = log ( 3log 10) ( 11log 10) = log
24 7.5 Exponential and Logarithmic Equations Solving an Exponen.al Equa.on Common Base Any equa+on that contains the form b cx, such as a = b cx, where the exponent includes a variable, is an exponen+al equa+on. If possible, rewrite each side as an expression of a common base raised to an exponent. Examples: What is the solu+on of 16 3x = 8? 16 3x = 8 (2 4 ) 3x = x = x = 3 x = ¼ What is the solu+on of 27 3x = 81? 27 3x = 81 (3 3 ) 3x = x = 3 4 9x = 4 x = 4 / 9
25 7.5 Exponential and Logarithmic Equations Solving an Exponen.al Equa.on Different Bases When the bases are not the same, you can take the logarithm of each side of the equa+on. If m and n are posi+ve, and m = n, then log m = log n. Examples: What is the solu+on of 15 3x = 285? 15 3x = 285 log15 3x = log285 (3x)log15 = log285 x What is the solu+on of 5 2x = 130? 5 2x = 130 ln5 2x = log130 (2x)ln5 = log130 x
26 7.5 Exponential and Logarithmic Equations Solving an Exponen.al Equa.on With a Graph What is the solu+on for 4 3x = 6000? Graph each half of the equa+on separately: Y 1 = 4 3x Y 2 = 6000 Adjust the window to view the point of intersec+on. The solu+on is x 2.09 What is the solu+on for 7 4x = 800? Y 1 = 7 4x Y 2 = 800 The solu+on is x
27 7.5 Exponential and Logarithmic Equations Modeling with an Exponen.al Func.on Wood is a sustainable, renewable, natural resource when you manage forests properly. Your lumber company has 1,200,000 trees. You plan to harvest 7% of the trees each year. How many years will it take to harvest half of the trees? T(n) = a = r = 7% = 0.07 b = 1 + r = 1 + ( 0.07) = 0.93 n =? It will take about 9.55 years to harvest half of the original trees. T(n) = ab n = (0.93) n 0.5 = 0.93 n ln(0.5) = ln(0.93 n ) ln(0.5) =n ln(0.93) 9.55 n
28 7.5 Exponential and Logarithmic Equations Solving a Logarithmic Equa.on A logarithmic equa+on is an equa+on that includes one or more logarithms involving a variable. Examples: What is the solu+on of log(4x 3) = 2? log(4x 3) = 2 10 log(4x 3) = x 3 = 100 4x = 103 x = 103/4 = What is the solu+on of ln (3 2x) = 1? ln (3 2x) = 1 e ln(3 2x) = e 1 3 2x = e 1 2x = e 1 3 x = (e 1 3)/
29 7.5 Exponential and Logarithmic Equations Using Logarithmic Proper.es to Solve an Equa.on What is the solu+on of log (x 3) + log (x) = 1? log(x 3) + log(x) = 1 log ((x 3)x) = 1 log (x 2 3x) = 1 10 log(x2 3x) = 10 1 x 2 3x = 10 x 2 3x 10 = 0 (x 5) (x + 2) = 0 x = 2, 5 Check:? log( 2 3) + log( 2) = 1? log(5 3) + log(5) = 1? log(2) + log(5) = = 1 x = 5
30 7.6 Natural Logarithms Simplifying a Natural Logarithmic Expression The func+on y = e x has an inverse, the natural logarithmic func+on y = log e x, or y = ln x. Example: What is 2ln 15 ln 17 wriuen as a single natural logarithm? 2ln 15 ln 17 = ln 15 2 ln 17 = ln 3
31 7.6 Natural Logarithms Solving a Natural Logarithmic Equa.on Use the inverse rela+onship between the func+ons y = ln x and y = e x to solve certain logarithmic and exponen+al equa+ons. Example: What are the solu+ons of ln (x 3) 2 = 4? ln (x 3) 2 = 4 e ln(x 3)2 = e 4 (x 3) 2 = e 4 x 3 = ±e 2 x = 3 ± e 2 x 10.39, 4.39 Check: ln ( ) 2? = 4 ln ( ) 2? =
32 7.6 Natural Logarithms Solving an Exponen.al Equa.on What is the solu+on of 4e 2x + 2 = 16? 4e 2x + 2 = 16 4e 2x = 14 e 2x = 3.5 ln e 2x = ln 3.5 2x lne = ln 3.5 x = (ln 3.5) / 2 x What is the solu+on of e 3x + 5 = 15? e 3x + 5 = 15 e 3x = 10 ln e 3x = ln 10 3x lne = ln 10 x = (ln 10) / 3 x 0.77
33 7.6 Natural Logarithms Using Natural Logarithms A spacecray can auain a stable orbit 300km above Earth if it reaches a velocity of 7.7 km/s. The formula for a rocket s maximum velocity v in kilometers per second is v = t + c lnr. The booster rocket fires for t seconds and the velocity of the exhaust is c km/s. The ra+o of the mass of the rocket filled with fuel to its mass without fuel is R. Suppose a rocket with exhaust velocity 2.8km/s has a mass ra+o of 25, and a firing +me of 100s. Can the spacecray auain a stable orbit 300km above Earth? R = 25, c = 2.8, t = 100 v = (100) + (2.8) ln(25) v (3.219) v 8.0 km/s Since 8.0km/s > 7.7km/s, the rocket can auain a stable orbit 300km above Earth
An equation of the form y = ab x where a 0 and the base b is a positive. x-axis (equation: y = 0) set of all real numbers
Algebra 2 Notes Section 7.1: Graph Exponential Growth Functions Objective(s): To graph and use exponential growth functions. Vocabulary: I. Exponential Function: An equation of the form y = ab x where
More information7-1. Exploring Exponential Models. Vocabulary. Review. Vocabulary Builder. Use Your Vocabulary. 1. Cross out the expressions that are NOT powers.
7-1 Eploring Eponential Models Vocabular Review 1. Cross out the epressions that are NOT powers. 16 6a 1 7. Circle the eponents in the epressions below. 5 6 5a z Vocabular Builder eponential deca (noun)
More informationPractice 6-1: Exponential Equations
Name Class Date Practice 6-1: Exponential Equations Which of the following are exponential functions? For those that are exponential functions, state the initial value and the base. For those that are
More information/8= Unit 8: ExPonential! Logs -4 2"" 1/16=
Unit 8: ExPonential! Logs Carli Castellano, Caroline Chivily, Julia Ashley Chapter 7 Exponential and logarithmic Functjons Section 7.1: Exploring Exponential Models Definitions An exponential function
More informationFLC Ch 9. Ex 2 Graph each function. Label at least 3 points and include any pertinent information (e.g. asymptotes). a) (# 14) b) (# 18) c) (# 24)
Math 5 Trigonometry Sec 9.: Exponential Functions Properties of Exponents a = b > 0, b the following statements are true: b x is a unique real number for all real numbers x f(x) = b x is a function with
More informationPRECAL REVIEW DAY 11/14/17
PRECAL REVIEW DAY 11/14/17 COPY THE FOLLOWING INTO JOURNAL 1 of 3 Transformations of logs Vertical Transformation Horizontal Transformation g x = log b x + c g x = log b x c g x = log b (x + c) g x = log
More informationSection 4.2 Logarithmic Functions & Applications
34 Section 4.2 Logarithmic Functions & Applications Recall that exponential functions are one-to-one since every horizontal line passes through at most one point on the graph of y = b x. So, an exponential
More informationExponential and Logarithmic Functions
Exponential and Logarithmic Functions Learning Targets 1. I can evaluate, analyze, and graph exponential functions. 2. I can solve problems involving exponential growth & decay. 3. I can evaluate expressions
More information8-1 Exploring Exponential Models
8- Eploring Eponential Models Eponential Function A function with the general form, where is a real number, a 0, b > 0 and b. Eample: y = 4() Growth Factor When b >, b is the growth factor Eample: y =
More informationUnit 3: Ra.onal and Radical Expressions. 3.1 Product Rule M1 5.8, M , M , 6.5,8. Objec.ve. Vocabulary o Base. o Scien.fic Nota.
Unit 3: Ra.onal and Radical Expressions M1 5.8, M2 10.1-4, M3 5.4-5, 6.5,8 Objec.ve 3.1 Product Rule I will be able to mul.ply powers when they have the same base, including simplifying algebraic expressions
More informationChapter 11 Logarithms
Chapter 11 Logarithms Lesson 1: Introduction to Logs Lesson 2: Graphs of Logs Lesson 3: The Natural Log Lesson 4: Log Laws Lesson 5: Equations of Logs using Log Laws Lesson 6: Exponential Equations using
More informationLesson 7 Practice Problems
Name: Date: Lesson 7 Section 7.1: Introduction to Logarithms 1. Locate the LOG button on your calculator. Use it to fill in the missing values in the input/output table. When you use your calculator, remember
More informationMath 180 Chapter 4 Lecture Notes. Professor Miguel Ornelas
Math 80 Chapter 4 Lecture Notes Professor Miguel Ornelas M. Ornelas Math 80 Lecture Notes Section 4. Section 4. Inverse Functions Definition of One-to-One Function A function f with domain D and range
More informationSection 4.4 Logarithmic and Exponential Equations
Section 4.4 Logarithmic and Exponential Equations Exponential Equations An exponential equation is one in which the variable occurs in the exponent. EXAMPLE: Solve the equation 2 x = 7. Solution 1: We
More informationStudy Guide and Review - Chapter 7
Choose a word or term from the list above that best completes each statement or phrase. 1. A function of the form f (x) = b x where b > 1 is a(n) function. exponential growth 2. In x = b y, the variable
More informationYou identified, graphed, and described several parent functions. (Lesson 1-5)
You identified, graphed, and described several parent functions. (Lesson 1-5) Evaluate, analyze, and graph exponential functions. Solve problems involving exponential growth and decay. algebraic function
More informationName Date Per. Ms. Williams/Mrs. Hertel
Name Date Per. Ms. Williams/Mrs. Hertel Day 7: Solving Exponential Word Problems involving Logarithms Warm Up Exponential growth occurs when a quantity increases by the same rate r in each period t. When
More informationDay Date Assignment. 7.1 Notes Exponential Growth and Decay HW: 7.1 Practice Packet Tuesday Wednesday Thursday Friday
1 Day Date Assignment Friday Monday /09/18 (A) /1/18 (B) 7.1 Notes Exponential Growth and Decay HW: 7.1 Practice Packet Tuesday Wednesday Thursday Friday Tuesday Wednesday Thursday Friday Monday /1/18
More informationGUIDED NOTES 6.4 GRAPHS OF LOGARITHMIC FUNCTIONS
GUIDED NOTES 6.4 GRAPHS OF LOGARITHMIC FUNCTIONS LEARNING OBJECTIVES In this section, you will: Identify the domain of a logarithmic function. Graph logarithmic functions. FINDING THE DOMAIN OF A LOGARITHMIC
More informationLesson 18 - Solving & Applying Exponential Equations Using Logarithms
Lesson 18 - Solving & Applying Exponential Equations Using Logarithms IB Math HL1 - Santowski 1 Fast Five! Solve the following:! (a) 5 x = 53! (b) log 3 38=x! (c) Solve 2 x = 7. HENCE, ALGEBRAICALLY solve
More information9.7 Common Logarithms, Natural Logarithms, and Change of Base
580 CHAPTER 9 Exponential and Logarithmic Functions Graph each function. 6. y = a x 2 b 7. y = 2 x + 8. y = log x 9. y = log / x Solve. 20. 2 x = 8 2. 9 = x -5 22. 4 x - = 8 x +2 2. 25 x = 25 x - 24. log
More informationSkill 6 Exponential and Logarithmic Functions
Skill 6 Exponential and Logarithmic Functions Skill 6a: Graphs of Exponential Functions Skill 6b: Solving Exponential Equations (not requiring logarithms) Skill 6c: Definition of Logarithms Skill 6d: Graphs
More information#2. Be able to identify what an exponential decay equation/function looks like.
1 Pre-AP Algebra II Chapter 7 Test Review Standards/Goals: G.2.a.: I can graph exponential and logarithmic functions with and without technology. G.2.b.: I can convert exponential equations to logarithmic
More informationGraphing Exponentials 6.0 Topic: Graphing Growth and Decay Functions
Graphing Exponentials 6.0 Topic: Graphing Growth and Decay Functions Date: Objectives: SWBAT (Graph Exponential Functions) Main Ideas: Mother Function Exponential Assignment: Parent Function: f(x) = b
More informationf(x) = d(x) q(x) + r(x).
Section 5.4: Dividing Polynomials 1. The division algorithm states, given a polynomial dividend, f(x), and non-zero polynomial divisor, d(x), where the degree of d(x) is less than or equal to the degree
More informationLogarithmic Functions
Logarithmic Functions Definition 1. For x > 0, a > 0, and a 1, y = log a x if and only if x = a y. The function f(x) = log a x is called the logarithmic function with base a. Example 1. Evaluate the following
More informationExponential and Logarithmic Functions
Graduate T.A. Department of Mathematics Dynamical Systems and Chaos San Diego State University April 9, 11 Definition (Exponential Function) An exponential function with base a is a function of the form
More informationChapter 3 Exponential and Logarithmic Functions
Chapter 3 Exponential and Logarithmic Functions Overview: 3.1 Exponential Functions and Their Graphs 3.2 Logarithmic Functions and Their Graphs 3.3 Properties of Logarithms 3.4 Solving Exponential and
More informationWrite each expression as a sum or difference of logarithms. All variables are positive. 4) log ( ) 843 6) Solve for x: 8 2x+3 = 467
Write each expression as a single logarithm: 10 Name Period 1) 2 log 6 - ½ log 9 + log 5 2) 4 ln 2 - ¾ ln 16 Write each expression as a sum or difference of logarithms. All variables are positive. 3) ln
More informationUnit 1 Study Guide Answers. 1a. Domain: 2, -3 Range: -3, 4, -4, 0 Inverse: {(-3,2), (4, -3), (-4, 2), (0, -3)}
Unit 1 Study Guide Answers 1a. Domain: 2, -3 Range: -3, 4, -4, 0 Inverse: {(-3,2), (4, -3), (-4, 2), (0, -3)} 1b. x 2-3 2-3 y -3 4-4 0 1c. no 2a. y = x 2b. y = mx+ b 2c. 2e. 2d. all real numbers 2f. yes
More information8.1 Apply Exponent Properties Involving Products. Learning Outcome To use properties of exponents involving products
8.1 Apply Exponent Properties Involving Products Learning Outcome To use properties of exponents involving products Product of Powers Property Let a be a real number, and let m and n be positive integers.
More informationPart 4: Exponential and Logarithmic Functions
Part 4: Exponential and Logarithmic Functions Chapter 5 I. Exponential Functions (5.1) II. The Natural Exponential Function (5.2) III. Logarithmic Functions (5.3) IV. Properties of Logarithms (5.4) V.
More informationEvaluate the exponential function at the specified value of x. 1) y = 4x, x = 3. 2) y = 2x, x = -3. 3) y = 243x, x = ) y = 16x, x = -0.
MAT 205-01C TEST 4 REVIEW (CHAP 13) NAME Evaluate the exponential function at the specified value of x. 1) y = 4x, x = 3 2) y = 2x, x = -3 3) y = 243x, x = 0.2 4) y = 16x, x = -0.25 Solve. 5) The number
More informationAlgebra 2 - Classwork April 25, Review
Name: ID: A Algebra 2 - Classwork April 25, 204 - Review Graph the exponential function.. y 4 x 2. Find the annual percent increase or decrease that y 0.5(2.) x models. a. 20% increase c. 5% decrease b.
More informationHW#1. Unit 4B Logarithmic Functions HW #1. 1) Which of the following is equivalent to y=log7 x? (1) y =x 7 (3) x = 7 y (2) x =y 7 (4) y =x 1/7
HW#1 Name Unit 4B Logarithmic Functions HW #1 Algebra II Mrs. Dailey 1) Which of the following is equivalent to y=log7 x? (1) y =x 7 (3) x = 7 y (2) x =y 7 (4) y =x 1/7 2) If the graph of y =6 x is reflected
More information4 Exponential and Logarithmic Functions
4 Exponential and Logarithmic Functions 4.1 Exponential Functions Definition 4.1 If a > 0 and a 1, then the exponential function with base a is given by fx) = a x. Examples: fx) = x, gx) = 10 x, hx) =
More informationExponential and Logarithmic Functions. 3. Pg #17-57 column; column and (need graph paper)
Algebra 2/Trig Unit 6 Notes Packet Name: Period: # Exponential and Logarithmic Functions 1. Worksheet 2. Worksheet 3. Pg 483-484 #17-57 column; 61-73 column and 76-77 (need graph paper) 4. Pg 483-484 #20-60
More informationChapter 3 Exponential and Logarithmic Functions
Chapter 3 Exponential and Logarithmic Functions Overview: 3.1 Exponential Functions and Their Graphs 3.2 Logarithmic Functions and Their Graphs 3.3 Properties of Logarithms 3.4 Solving Exponential and
More informationEXAM 3 Tuesday, March 18, 2003
MATH 12001 Precalculus: Algebra & Trigonometry Spring 2003 Sections 2 & 3 Darci L. Kracht Name: Score: /100. 115 pts available EXAM 3 Tuesday, March 18, 2003 Part I: NO CALCULATORS. (You must turn this
More informationMath 103 Intermediate Algebra Final Exam Review Practice Problems
Math 10 Intermediate Algebra Final Eam Review Practice Problems The final eam covers Chapter, Chapter, Sections 4.1 4., Chapter 5, Sections 6.1-6.4, 6.6-6.7, Chapter 7, Chapter 8, and Chapter 9. The list
More information2.6 Logarithmic Functions. Inverse Functions. Question: What is the relationship between f(x) = x 2 and g(x) = x?
Inverse Functions Question: What is the relationship between f(x) = x 3 and g(x) = 3 x? Question: What is the relationship between f(x) = x 2 and g(x) = x? Definition (One-to-One Function) A function f
More informationCHAPTER 5: Exponential and Logarithmic Functions
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 5: Exponential and Logarithmic Functions 5.1 Inverse Functions 5.2 Exponential Functions and Graphs 5.3 Logarithmic Functions
More information4.4 Graphs of Logarithmic Functions
590 Chapter 4 Exponential and Logarithmic Functions 4.4 Graphs of Logarithmic Functions In this section, you will: Learning Objectives 4.4.1 Identify the domain of a logarithmic function. 4.4.2 Graph logarithmic
More informationMATH 1113 Exam 2 Review. Fall 2017
MATH 1113 Exam 2 Review Fall 2017 Topics Covered Section 3.1: Inverse Functions Section 3.2: Exponential Functions Section 3.3: Logarithmic Functions Section 3.4: Properties of Logarithms Section 3.5:
More informationObjectives. Use the number e to write and graph exponential functions representing realworld
Objectives Use the number e to write and graph exponential functions representing realworld situations. Solve equations and problems involving e or natural logarithms. natural logarithm Vocabulary natural
More informationExponents and Logarithms Exam
Name: Class: Date: Exponents and Logarithms Exam Multiple Choice Identify the choice that best completes the statement or answers the question.. The decay of a mass of a radioactive sample can be represented
More informationCHAPTER FIVE. Solutions for Section 5.1. Skill Refresher. Exercises
CHAPTER FIVE 5.1 SOLUTIONS 265 Solutions for Section 5.1 Skill Refresher S1. Since 1,000,000 = 10 6, we have x = 6. S2. Since 0.01 = 10 2, we have t = 2. S3. Since e 3 = ( e 3) 1/2 = e 3/2, we have z =
More informationChapter 7 Exponential and Logarithmic Functions Review Packet
Non-Calculator Chapter 7 Exponential and Logarithmic Functions Review Packet Possible topics: Graphing exponential and logarithmic functions (and their transformations), switching between logarithmic and
More informationIntermediate Algebra Chapter 12 Review
Intermediate Algebra Chapter 1 Review Set up a Table of Coordinates and graph the given functions. Find the y-intercept. Label at least three points on the graph. Your graph must have the correct shape.
More informationWhat You Need to Know for the Chapter 7 Test
Score: /46 Name: Date: / / Hr: Alg 2C Chapter 7 Review - WYNTK CH 7 What You Need to Know for the Chapter 7 Test 7.1 Write & evaluate exponential expressions to model growth and decay situations. Determine
More informationMAC Module 8 Exponential and Logarithmic Functions I. Rev.S08
MAC 1105 Module 8 Exponential and Logarithmic Functions I Learning Objectives Upon completing this module, you should be able to: 1. Distinguish between linear and exponential growth. 2. Model data with
More informationMAC Module 8. Exponential and Logarithmic Functions I. Learning Objectives. - Exponential Functions - Logarithmic Functions
MAC 1105 Module 8 Exponential and Logarithmic Functions I Learning Objectives Upon completing this module, you should be able to: 1. Distinguish between linear and exponential growth. 2. Model data with
More informationGrowth 23%
y 100 0. 4 x Decay 23% Math 109C - Fall 2012 page 16 39. Write the quantity 12,600,000,000 miles in scientific notation. The result is: (A) 12. 6 x 10 9 miles (B) 12. 6 x 10 9 miles (C) 1. 26 x 10 10 miles
More informationCHAPTER 7. Logarithmic Functions
CHAPTER 7 Logarithmic Functions 7.1 CHARACTERISTICS OF LOGARITHMIC FUNCTIONS WITH BASE 10 AND BASE E Chapter 7 LOGARITHMS Logarithms are a new operation that we will learn. Similar to exponential functions,
More information, identify what the letters P, r, n and t stand for.
1.In the formula At p 1 r n nt, identify what the letters P, r, n and t stand for. 2. Find the exponential function whose graph is given f(x) = a x 3. State the domain and range of the function (Enter
More informationChapter 6: Exponential and Logarithmic Functions
Section 6.1: Algebra and Composition of Functions #1-9: Let f(x) = 2x + 3 and g(x) = 3 x. Find each function. 1) (f + g)(x) 2) (g f)(x) 3) (f/g)(x) 4) ( )( ) 5) ( g/f)(x) 6) ( )( ) 7) ( )( ) 8) (g+f)(x)
More informationComposition of Functions
Math 120 Intermediate Algebra Sec 9.1: Composite and Inverse Functions Composition of Functions The composite function f g, the composition of f and g, is defined as (f g)(x) = f(g(x)). Recall that a function
More informationHonors Advanced Algebra Chapter 8 Exponential and Logarithmic Functions and Relations Target Goals
Honors Advanced Algebra Chapter 8 Exponential and Logarithmic Functions and Relations Target Goals By the end of this chapter, you should be able to Graph exponential growth functions. (8.1) Graph exponential
More informationSkill 6 Exponential and Logarithmic Functions
Skill 6 Exponential and Logarithmic Functions Skill 6a: Graphs of Exponential Functions Skill 6b: Solving Exponential Equations (not requiring logarithms) Skill 6c: Definition of Logarithms Skill 6d: Graphs
More informationMath M110: Lecture Notes For Chapter 12 Section 12.1: Inverse and Composite Functions
Math M110: Lecture Notes For Chapter 12 Section 12.1: Inverse and Composite Functions Inverse function (interchange x and y): Find the equation of the inverses for: y = 2x + 5 ; y = x 2 + 4 Function: (Vertical
More informationGraphing Quadratic Functions 9.1
Quadratic Functions - Graphing Quadratic Functions 9.1 f ( x) = a x + b x + c (also called standard form). The graph of quadratic functions is called a parabola. Axis of Symmetry a central line which makes
More informationMATH 1113 Exam 2 Review. Spring 2018
MATH 1113 Exam 2 Review Spring 2018 Section 3.1: Inverse Functions Topics Covered Section 3.2: Exponential Functions Section 3.3: Logarithmic Functions Section 3.4: Properties of Logarithms Section 3.5:
More informationExample. Determine the inverse of the given function (if it exists). f(x) = 3
Example. Determine the inverse of the given function (if it exists). f(x) = g(x) = p x + x We know want to look at two di erent types of functions, called logarithmic functions and exponential functions.
More information4.1 Exponential Functions
Chapter 4 Exponential and Logarithmic Functions 531 4.1 Exponential Functions In this section, you will: Learning Objectives 4.1.1 Evaluate exponential functions. 4.1.2 Find the equation of an exponential
More informationO5C1: Graphing Exponential Functions
Name: Class Period: Date: Algebra 2 Honors O5C1-4 REVIEW O5C1: Graphing Exponential Functions Graph the exponential function and fill in the table to the right. You will need to draw in the x- and y- axis.
More informationExponential and Logarithmic Functions. Copyright Cengage Learning. All rights reserved.
3 Exponential and Logarithmic Functions Copyright Cengage Learning. All rights reserved. 3.1 Exponential Functions and Their Graphs Copyright Cengage Learning. All rights reserved. What You Should Learn
More informationMath 095 Final Exam Review - MLC
Math 095 Final Exam Review - MLC Although this is a comprehensive review, you should also look over your old reviews from previous modules, the readings, and your notes. Round to the thousandth unless
More information7.1 Exponential Functions
7.1 Exponential Functions 1. What is 16 3/2? Definition of Exponential Functions Question. What is 2 2? Theorem. To evaluate a b, when b is irrational (so b is not a fraction of integers), we approximate
More informationAnother enormous super-family of functions are exponential functions.
Hartfield College Algebra (Version 2018 - Thomas Hartfield) Unit FIVE Page - 1 - of 39 Topic 37: Exponential Functions In previous topics we ve discussed power functions, n functions of the form f x x,
More informationSection 2.3: Logarithmic Functions Lecture 3 MTH 124
Procedural Skills Learning Objectives 1. Build an exponential function using the correct compounding identifiers (annually, monthly, continuously etc...) 2. Manipulate exponents algebraically. e.g. Solving
More informationThe Exponential function f with base b is f (x) = b x where b > 0, b 1, x a real number
Chapter 4: 4.1: Exponential Functions Definition: Graphs of y = b x Exponential and Logarithmic Functions The Exponential function f with base b is f (x) = b x where b > 0, b 1, x a real number Graph:
More informationReview of Functions A relation is a function if each input has exactly output. The graph of a function passes the vertical line test.
CA-Fall 011-Jordan College Algebra, 4 th edition, Beecher/Penna/Bittinger, Pearson/Addison Wesley, 01 Chapter 5: Exponential Functions and Logarithmic Functions 1 Section 5.1 Inverse Functions Inverse
More informationf(x) = 2x + 5 3x 1. f 1 (x) = x + 5 3x 2. f(x) = 102x x
1. Let f(x) = x 3 + 7x 2 x 2. Use the fact that f( 1) = 0 to factor f completely. (2x-1)(3x+2)(x+1). 2. Find x if log 2 x = 5. x = 1/32 3. Find the vertex of the parabola given by f(x) = 2x 2 + 3x 4. (Give
More informationfor every x in the gomain of g
Section.7 Definition of Inverse Function Let f and g be two functions such that f(g(x)) = x for every x in the gomain of g and g(f(x)) = x for every x in the gomain of f Under these conditions, the function
More information6.3 logarithmic FUnCTIOnS
SECTION 6.3 logarithmic functions 4 9 1 learning ObjeCTIveS In this section, you will: Convert from logarithmic to exponential form. Convert from exponential to logarithmic form. Evaluate logarithms. Use
More informationMATH 1431-Precalculus I
MATH 43-Precalculus I Chapter 4- (Composition, Inverse), Eponential, Logarithmic Functions I. Composition of a Function/Composite Function A. Definition: Combining of functions that output of one function
More informationUnit 4 Rational Expressions. Mrs. Valen+ne Math III
Unit 4 Rational Expressions Mrs. Valen+ne Math III 4.1 Simplifying Rational Expressions Simplifying Rational Expressions Expression in the form Simplifying a rational expression is like simplifying any
More informationSection Exponential Functions
121 Section 4.1 - Exponential Functions Exponential functions are extremely important in both economics and science. It allows us to discuss the growth of money in a money market account as well as the
More informationConcept Category 2. Exponential and Log Functions
Concept Category 2 Exponential and Log Functions Concept Category 2 Check List *Find the inverse and composition of functions *Identify an exponential from a table, graph and equation *Identify the difference
More informationMATH 1101 Exam 3 Review - Gentry. Spring 2018
MATH 1101 Exam 3 Review - Gentry Spring 2018 Topics Covered Section 5.3 Fitting Exponential Functions to Data Section 5.4 Logarithmic Functions Section 5.5 Modeling with Logarithmic Functions What s in
More informationMath 1101 Exam 3 Practice Problems
Math 1101 Exam 3 Practice Problems These problems are not intended to cover all possible test topics. These problems should serve as an activity in preparing for your test, but other study is required
More informationMath 112 Fall 2015 Midterm 2 Review Problems Page 1. has a maximum or minimum and then determine the maximum or minimum value.
Math Fall 05 Midterm Review Problems Page f 84 00 has a maimum or minimum and then determine the maimum or minimum value.. Determine whether Ma = 00 Min = 00 Min = 8 Ma = 5 (E) Ma = 84. Consider the function
More informationFinal Exam Study Aid
Math 112 Final Exam Study Aid 1 of 33 Final Exam Study Aid Note: This study aid is intended to help you review for the final exam. It covers the primary concepts in the course, with a large emphasis on
More information2015 2nd Semester Exam Review
Algebra 2 2015 2nd Semester Exam Review 1. Write a function whose graph is a translation of the graph of the function in two directions. Describe the translation. 2. What are the solutions to the equation?
More informationGOOD LUCK! 2. a b c d e 12. a b c d e. 3. a b c d e 13. a b c d e. 4. a b c d e 14. a b c d e. 5. a b c d e 15. a b c d e. 6. a b c d e 16.
MA109 College Algebra Fall 2018 Practice Final Exam 2018-12-12 Name: Sec.: Do not remove this answer page you will turn in the entire exam. You have two hours to do this exam. No books or notes may be
More informationTeacher: Mr. Chafayay. Name: Class & Block : Date: ID: A. 3 Which function is represented by the graph?
Teacher: Mr hafayay Name: lass & lock : ate: I: Midterm Exam Math III H Multiple hoice Identify the choice that best completes the statement or answers the question Which function is represented by the
More informationName. 6) f(x) = x Find the inverse of the given function. 1) f(x) = x + 5. Evaluate. 7) Let g(x) = 6x. Find g(3) 2) f(x) = -3x
Exam 2 Preparation Ch 5 & 6 v01 There will be 25 questions on Exam 2. Fourteen questions from chapter 5 and eleven questions from chapter 6. No Book/No Notes/No Ipod/ No Phone/Yes Calculator/55 minutes
More information1 Functions, Graphs and Limits
1 Functions, Graphs and Limits 1.1 The Cartesian Plane In this course we will be dealing a lot with the Cartesian plane (also called the xy-plane), so this section should serve as a review of it and its
More informationA. Evaluate log Evaluate Logarithms
A. Evaluate log 2 16. Evaluate Logarithms Evaluate Logarithms B. Evaluate. C. Evaluate. Evaluate Logarithms D. Evaluate log 17 17. Evaluate Logarithms Evaluate. A. 4 B. 4 C. 2 D. 2 A. Evaluate log 8 512.
More informationMATH 1113 Exam 2 Review
MATH 1113 Exam 2 Review Section 3.1: Inverse Functions Topics Covered Section 3.2: Exponential Functions Section 3.3: Logarithmic Functions Section 3.4: Properties of Logarithms Section 3.5: Exponential
More informationLogarithmic Functions and Models Power Functions Logistic Function. Mathematics. Rosella Castellano. Rome, University of Tor Vergata
Mathematics Rome, University of Tor Vergata The logarithm is used to model real-world phenomena in numerous elds: i.e physics, nance, economics, etc. From the equation 4 2 = 16 we see that the power to
More informationMATH 1101 Chapter 5 Review
MATH 1101 Chapter 5 Review Section 5.1 Exponential Growth Functions Section 5.2 Exponential Decay Functions Topics Covered Section 5.3 Fitting Exponential Functions to Data Section 5.4 Logarithmic Functions
More informationMAC Learning Objectives. Logarithmic Functions. Module 8 Logarithmic Functions
MAC 1140 Module 8 Logarithmic Functions Learning Objectives Upon completing this module, you should be able to 1. evaluate the common logarithmic function. 2. solve basic exponential and logarithmic equations.
More information4. Sketch the graph of the function. Ans: A 9. Sketch the graph of the function. Ans B. Version 1 Page 1
Name: Online ECh5 Prep Date: Scientific Calc ONLY! 4. Sketch the graph of the function. A) 9. Sketch the graph of the function. B) Ans B Version 1 Page 1 _ 10. Use a graphing utility to determine which
More informationFinal Exam Review: Study Guide Math 3
Final Exam Review: Study Guide Math 3 Name: Day 1 Functions, Graphing, Regression Relation: Function: Domain: Range: Asymptote: Hole: Graphs of Functions f(x) = x f(x) = f(x) = x f(x) = x 3 Key Ideas Key
More informationMATH 1040 CP 11 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
MATH 1040 CP 11 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Write the equation in its equivalent exponential form. 1) log 5 125 = 3 1) 2) log 2 16
More informationExponential and Logarithmic Functions
C H A P T ER Exponential and Logarithmic Functions Scarlet macaws are native to the jungles of Southern Mexico and Central America, and can live up to 75 years. However, macaws and other birds are threatened
More informationExponential and Logarithmic Functions. Copyright Cengage Learning. All rights reserved.
Exponential and Logarithmic Functions Copyright Cengage Learning. All rights reserved. 4.6 Modeling With Exponential And Logarithmic Functions Copyright Cengage Learning. All rights reserved. Objectives
More informationGeometry Placement Exam Review Revised 2017 Maine East High School
Geometry Placement Exam Review Revised 017 Maine East High School The actual placement exam has 91 questions. The placement exam is free response students must solve questions and write answer in space
More informationSec. 4.2 Logarithmic Functions
Sec. 4.2 Logarithmic Functions The Logarithmic Function with Base a has domain all positive real numbers and is defined by Where and is the inverse function of So and Logarithms are inverses of Exponential
More information