Exponential and Logarithmic Functions
|
|
- Garry Lewis
- 5 years ago
- Views:
Transcription
1 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 involving logarithms.
2 What is different about the two functions below? f(x) = x 3 g(x) = 3 x
3 What is different about the two functions below? f(x) = x 3 This is an Algebraic Function, which is a function whose values are obtained by multiplying, dividing, adding, and/or subtracting constants and the independent variable, or raising the independent variable to a natural power. g(x) = 3 x This is an Exponential Function, which is a function whose base is a constant and the exponent is a variable. Exponential and Logarithmic Functions are considered,because they cannot be expressed in terms of algebraic operations.
4 An with base b has the form f(x) = ab x, where: x is any real number, a and b are real number constants a 0 b is positive, and b 1 Examples: Non-Examples:
5 1. Sketch and analyze the graph of the function. Describe its Domain, Range, intercepts, asymptotes, end-behavior, and where the function is increasing or decreasing. f(x) = 4 x
6 2. Sketch and analyze the graph of the function. Describe its Domain, Range, intercepts, asymptotes, end-behavior, and where the function is increasing or decreasing. f(x) = 5 x
7 3. Use the graph of f(x) = (½) x to describe the transformation that results in the given function. Then sketch the graph of the function. g(x) = (½) x + 1
8 4. Use the graph of f(x) = (½) x to describe the transformation that results in the given function. Then sketch the graph of the function. h(x) = (½) x
9 5. Use the graph of f(x) = (½) x to describe the transformation that results in the given function. Then sketch the graph of the function. j(x) = -2(½) x
10 In most real-world applications involving exponential functions, the most commonly used base is not 2 or 10, but an irrational number e called the. e = The value of e is equal to the limit of approaches infinity. (1 1 x )x as x
11 Graph of f(x) = e x Exponential Functions
12 6. Use the graph of f(x) = e x to describe the transformation that results in the graph of each function. g(x) = e 3x h(x) = e x 1 j(x) = 2e x
13 Compound Interest Formula If a principal amount P is invested at an annual interest rate r (in decimal form) compounded n times a year, then the balance A in the account after t years is given by A P(1 r n )nt
14 7. Mrs. Saliman invested $2000 into an educational account for her daughter when she was an infant. The account has a 5% interest rate. If Mrs. Saliman does not make any other deposits or withdrawals, what will the account balance be after 18 years if the interest is compounded: a. quarterly? b. monthly? c. daily?
15 Suppose the interest were compounded continuously so that there was no waiting period between interest payments. We can derive a formula for. If a principal P is invested at an annual rate r (in decimal form) compounded continuously, then the balance A in the account after t years is given by A Pe rt
16 8. Mrs. Saliman found an account that will pay the 5% interest compounded continuously on her $2000 educational investment. What will be her account balance after 18 years?
17 If an initial quantity N 0 grows or decays at an exponential rate r or k (as a decimal), then the final amount N after a time t is given by the following formulas. Exponential Growth or Decay N N 0 (1 r) t If r is a growth rate, r > 0 If r is a decay rate, r < 0 Continuous Exponential Growth or Decay N N 0 e kt If k is a continuous growth rate, then k > 0 If k is a continuous decay rate, then k < 0
18 9. A state s population is declining at a rate of 2.6% annually. The state currently has a population of approximately 11 million people. If the population continues to decline at this, predict the population of the state in 30 years if the rate of decay is 2.6% annually and continuously.
19 Logarithmic Functions The inverse of f(x) = b x is called a with base b, denoted log b x and read log base b of x.
20 Logarithmic Functions Logarithmic Form log b x y Exponential Form Therefore, b y x b log b x x
21 Logarithmic Functions 1. Evaluate each logarithm log 2 16 a. b. log log 3 27 c. d. log 17 17
22 Logarithmic Functions 2. Evaluate each expression a. log b log
23 Logarithmic Functions A logarithm with base 10 or is called a, and is often written without the base. Basic Properties of Common Logarithms log1 0 log 10 log10 1 log10 x x 10 logx x
24 Logarithmic Functions 3. Evaluate each expression log10,000 a. b. 10 log12 log14 c. d. log( 11)
25 Logarithmic Functions A logarithm with base e or log e is called a and is denoted ln. Basic Properties of Natural Logarithms ln 1 = 0 ln e = 1 ln e x = x e ln x x
26 Logarithmic Functions 4. Evaluate each expression lne a. b. ln( 1.2) c. d. e ln 4 ln 7
27 Logarithmic Functions 5. Sketch and analyze the graph of the function. Describe its domain, range, intercepts, asymptotes, end behavior, and where the function is increasing or decreasing. f (x) log 2 x
28 Logarithmic Functions 6. Use the graph of f(x) = log x to describe the transformation that results in each function. a. g(x) = log (x+1) b. h(x) = log(x+2) c. j(x) = 5 log (x-3)
4 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 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 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 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 information5.1. EXPONENTIAL FUNCTIONS AND THEIR GRAPHS
5.1. EXPONENTIAL FUNCTIONS AND THEIR GRAPHS 1 What You Should Learn Recognize and evaluate exponential functions with base a. Graph exponential functions and use the One-to-One Property. Recognize, evaluate,
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 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 informationExponential Functions and Their Graphs (Section 3-1)
Exponential Functions and Their Graphs (Section 3-1) Essential Question: How do you graph an exponential function? Students will write a summary describing the steps for graphing an exponential function.
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 information3-1: Exponential Functions
3-1: Exponeial Functions Precalculus Mr. Gallo Exponeial Function Types: Exponeial Growth as x increases, y increases Exponeial Decay as x increases, y decreases approaching zero Asymptote Line the graph
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 informationMock Final Exam Name. Solve and check the linear equation. 1) (-8x + 8) + 1 = -7(x + 3) A) {- 30} B) {- 6} C) {30} D) {- 28}
Mock Final Exam Name Solve and check the linear equation. 1) (-8x + 8) + 1 = -7(x + 3) 1) A) {- 30} B) {- 6} C) {30} D) {- 28} First, write the value(s) that make the denominator(s) zero. Then solve the
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 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 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 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 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 informationnt and A = Pe rt to solve. 3) Find the accumulated value of an investment of $10,000 at 4% compounded semiannually for 5 years.
Exam 4 Review Approximate the number using a calculator. Round your answer to three decimal places. 1) 2 1.7 2) e -1.4 Use the compound interest formulas A = P 1 + r n nt and A = Pe rt to solve. 3) Find
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 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 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 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 informationAn 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 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 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 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 information( ) ( ) x. The exponential function f(x) with base b is denoted by x
Page of 7 Eponential and Logarithmic Functions Eponential Functions and Their Graphs: Section Objectives: Students will know how to recognize, graph, and evaluate eponential functions. The eponential function
More informationOBJECTIVE 4 EXPONENTIAL FORM SHAPE OF 5/19/2016. An exponential function is a function of the form. where b > 0 and b 1. Exponential & Log Functions
OBJECTIVE 4 Eponential & Log Functions EXPONENTIAL FORM An eponential function is a function of the form where > 0 and. f ( ) SHAPE OF > increasing 0 < < decreasing PROPERTIES OF THE BASIC EXPONENTIAL
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 informationGUIDED NOTES 6.1 EXPONENTIAL FUNCTIONS
GUIDED NOTES 6.1 EXPONENTIAL FUNCTIONS LEARNING OBJECTIVES In this section, you will: Evaluate exponential functions. Find the equation of an exponential function. Use compound interest formulas. Evaluate
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 informationEvaluate the expression using the values given in the table. 1) (f g)(6) x f(x) x g(x)
M60 (Precalculus) ch5 practice test Evaluate the expression using the values given in the table. 1) (f g)(6) 1) x 1 4 8 1 f(x) -4 8 0 15 x -5-4 1 6 g(x) 1-5 4 8 For the given functions f and g, find the
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 informationLogarithmic, Exponential, and Other Transcendental Functions. Copyright Cengage Learning. All rights reserved.
5 Logarithmic, Exponential, and Other Transcendental Functions Copyright Cengage Learning. All rights reserved. 5.5 Bases Other Than e and Applications Copyright Cengage Learning. All rights reserved.
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 informationMA Lesson 14 Notes Summer 2016 Exponential Functions
Solving Eponential Equations: There are two strategies used for solving an eponential equation. The first strategy, if possible, is to write each side of the equation using the same base. 3 E : Solve:
More informationThe units on the average rate of change in this situation are. change, and we would expect the graph to be. ab where a 0 and b 0.
Lesson 9: Exponential Functions Outline Objectives: I can analyze and interpret the behavior of exponential functions. I can solve exponential equations analytically and graphically. I can determine the
More informationChapter 2 Functions and Graphs
Chapter 2 Functions and Graphs Section 5 Exponential Functions Objectives for Section 2.5 Exponential Functions The student will be able to graph and identify the properties of exponential functions. The
More informationC. HECKMAN TEST 1A SOLUTIONS 170
C. HECKMAN TEST 1A SOLUTIONS 170 1) Thornley s Bank of Atlanta offers savings accounts which earn 4.5% per year. You have $00, which you want to invest. a) [10 points] If the bank compounds the interest
More informationUnit 5: Exponential and Logarithmic Functions
71 Rational eponents Unit 5: Eponential and Logarithmic Functions If b is a real number and n and m are positive and have no common factors, then n m m b = b ( b ) m n n Laws of eponents a) b) c) d) e)
More informationGUIDED NOTES 6.1 EXPONENTIAL FUNCTIONS
GUIDED NOTES 6.1 EXPONENTIAL FUNCTIONS LEARNING OBJECTIVES In this section, you will: Evaluate exponential functions. Find the equation of an exponential function. Use compound interest formulas. Evaluate
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 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 informationThe function is defined for all values of x. Therefore, the domain is set of all real numbers.
Graph each function. State the domain and range. 1. f (x) = 3 x 3 + 2 The function is defined for all values of x. Therefore, the domain is set of all real numbers. The value of f (x) tends to 2 as x tends
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 informationAlgebra 32 Midterm Review Packet
Algebra 2 Midterm Review Packet Formulas you will receive on the Midterm: y = a b x A = Pe rt A = P (1 + r n ) nt A = P(1 + r) t A = P(1 r) t x = b ± b2 4ac 2a Name: Teacher: Day/Period: Date of Midterm:
More informationContinuously Compounded Interest. Simple Interest Growth. Simple Interest. Logarithms and Exponential Functions
Exponential Models Clues in the word problems tell you which formula to use. If there s no mention of compounding, use a growth or decay model. If your interest is compounded, check for the word continuous.
More informationLecture 7: Sections 2.3 and 2.4 Rational and Exponential Functions. Recall that a power function has the form f(x) = x r where r is a real number.
L7-1 Lecture 7: Sections 2.3 and 2.4 Rational and Exponential Functions Recall that a power function has the form f(x) = x r where r is a real number. f(x) = x 1/2 f(x) = x 1/3 ex. Sketch the graph of
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 informationINTERNET MAT 117 Review Problems. (1) Let us consider the circle with equation. (b) Find the center and the radius of the circle given above.
INTERNET MAT 117 Review Problems (1) Let us consider the circle with equation x 2 + y 2 + 2x + 3y + 3 4 = 0. (a) Find the standard form of the equation of the circle given above. (b) Find the center and
More informationTopic 33: One-to-One Functions. Are the following functions one-to-one over their domains?
Topic 33: One-to-One Functions Definition: A function f is said to be one-to-one if for every value f(x) in the range of f there is exactly one corresponding x-value in the domain of f. Ex. Are the following
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 informationInverse Functions. Definition 1. The exponential function f with base a is denoted by. f(x) = a x
Inverse Functions Definition 1. The exponential function f with base a is denoted by f(x) = a x where a > 0, a 1, and x is any real number. Example 1. In the same coordinate plane, sketch the graph of
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 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 information2(x 4 7x 2 18) 2(x 2 9)(x 2 + 2) 2(x 3)(x + 3)(x 2 + 2)
Completely factor 2x 4 14x 2 36 2(x 4 7x 2 18) 2(x 2 9)(x 2 + 2) 2(x 3)(x + 3)(x 2 + 2) Add and simplify Simplify as much as possible Subtract and simplify Determine the inverse of Multiply and simplify
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 informationFinal Exam C Name i D) 2. Solve the equation by factoring. 4) x2 = x + 72 A) {1, 72} B) {-8, 9} C) {-8, -9} D) {8, 9} 9 ± i
Final Exam C Name First, write the value(s) that make the denominator(s) zero. Then solve the equation. 7 ) x + + 3 x - = 6 (x + )(x - ) ) A) No restrictions; {} B) x -, ; C) x -; {} D) x -, ; {2} Add
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 informationHomework 3. (33-40) The graph of an exponential function is given. Match each graph to one of the following functions.
Homework Section 4. (-40) The graph of an exponential function is given. Match each graph to one of the following functions. (a)y = x (b)y = x (c)y = x (d)y = x (e)y = x (f)y = x (g)y = x (h)y = x (46,
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 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 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 informationFinal Exam A Name. 20 i C) Solve the equation by factoring. 4) x2 = x + 30 A) {-5, 6} B) {5, 6} C) {1, 30} D) {-5, -6} -9 ± i 3 14
Final Exam A Name First, write the value(s) that make the denominator(s) zero. Then solve the equation. 1 1) x + 3 + 5 x - 3 = 30 (x + 3)(x - 3) 1) A) x -3, 3; B) x -3, 3; {4} C) No restrictions; {3} D)
More informationAlgebra 2 & Trigonometry Honors Midterm Review 2016
Algebra & Trigonometry Honors Midterm Review 016 Solving Equations 1) Find all values of x that satisfy the equation, 5x 1 = x + 3 ) Solve the following by completing the square. Express your answer in
More informationExponential and Logarithmic Functions. Exponential Functions. Example. Example
Eponential and Logarithmic Functions Math 1404 Precalculus Eponential and 1 Eample Eample Suppose you are a salaried employee, that is, you are paid a fied sum each pay period no matter how many hours
More information10 Exponential and Logarithmic Functions
10 Exponential and Logarithmic Functions Concepts: Rules of Exponents Exponential Functions Power Functions vs. Exponential Functions The Definition of an Exponential Function Graphing Exponential Functions
More informationLogarithmic and Exponential Equations and Inequalities College Costs
Logarithmic and Exponential Equations and Inequalities ACTIVITY 2.6 SUGGESTED LEARNING STRATEGIES: Summarize/ Paraphrase/Retell, Create Representations Wesley is researching college costs. He is considering
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 informationMath M111: Lecture Notes For Chapter 10
Math M: Lecture Notes For Chapter 0 Sections 0.: Inverse Function Inverse function (interchange and y): Find the equation of the inverses for: y = + 5 ; y = + 4 3 Function (from section 3.5): (Vertical
More informationSummer MA Lesson 20 Section 2.7 (part 2), Section 4.1
Summer MA 500 Lesson 0 Section.7 (part ), Section 4. Definition of the Inverse of a Function: Let f and g be two functions such that f ( g ( )) for every in the domain of g and g( f( )) for every in the
More informationReview of Exponential Relations
Review of Exponential Relations Integrated Math 2 1 Concepts to Know From Video Notes/ HW & Lesson Notes Zero and Integer Exponents Exponent Laws Scientific Notation Analyzing Data Sets (M&M Lab & HW/video
More informationTransformations of Functions and Exponential Functions January 24, / 35
Exponential Functions January 24, 2017 Exponential Functions January 24, 2017 1 / 35 Review of Section 1.2 Reminder: Week-in-Review, Help Sessions, Oce Hours Mathematical Models Linear Regression Function
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 informationExponential Functions
Exponential Functions MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 2011 Objectives In this lesson we will learn to: recognize and evaluate exponential functions with base a,
More informationPractice Questions for Final Exam - Math 1060Q - Fall 2014
Practice Questions for Final Exam - Math 1060Q - Fall 01 Before anyone asks, the final exam is cumulative. It will consist of about 50% problems on exponential and logarithmic functions, 5% problems on
More informationINTERNET MAT 117. Solution for the Review Problems. (1) Let us consider the circle with equation. x 2 + 2x + y 2 + 3y = 3 4. (x + 1) 2 + (y + 3 2
INTERNET MAT 117 Solution for the Review Problems (1) Let us consider the circle with equation x 2 + y 2 + 2x + 3y + 3 4 = 0. (a) Find the standard form of the equation of the circle given above. (i) Group
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 informationAlgebra 2 Honors. Logs Test Review
Algebra 2 Honors Logs Test Review Name Date Let ( ) = ( ) = ( ) =. Perform the indicated operation and state the domain when necessary. 1. ( (6)) 2. ( ( 3)) 3. ( (6)) 4. ( ( )) 5. ( ( )) 6. ( ( )) 7. (
More informationPre-Calculus Final Exam Review Units 1-3
Pre-Calculus Final Exam Review Units 1-3 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Find the value for the function. Find f(x - 1) when f(x) = 3x
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 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 informationPolynomials and Rational Functions (2.1) The shape of the graph of a polynomial function is related to the degree of the polynomial
Polynomials and Rational Functions (2.1) The shape of the graph of a polynomial function is related to the degree of the polynomial Shapes of Polynomials Look at the shape of the odd degree polynomials
More information1. The dosage in milligrams D of a heartworm preventive for a dog who weighs X pounds is given by D x. Substitute 28 in place of x to get:
1. The dosage in milligrams D of a heartworm preventive for a dog who weighs X pounds is given by D x 28 pounds. ( ) = 136 ( ). Find the proper dosage for a dog that weighs 25 x Substitute 28 in place
More informationInternet Mat117 Formulas and Concepts. d(a, B) = (x 2 x 1 ) 2 + (y 2 y 1 ) 2. ( x 1 + x 2 2., y 1 + y 2. (x h) 2 + (y k) 2 = r 2. m = y 2 y 1 x 2 x 1
Internet Mat117 Formulas and Concepts 1. The distance between the points A(x 1, y 1 ) and B(x 2, y 2 ) in the plane is d(a, B) = (x 2 x 1 ) 2 + (y 2 y 1 ) 2. 2. The midpoint of the line segment from A(x
More informationAlgebra III: Blizzard Bag #1 Exponential and Logarithm Functions
NAME : DATE: PERIOD: Algebra III: Blizzard Bag #1 Exponential and Logarithm Functions Students need to complete the following assignment, which will aid in review for the end of course exam. Look back
More informationMath 101 Final Exam Review Solutions. Eric Schmutz
Math 101 Final Exam Review Solutions Eric Schmutz Problem 1. Write an equation of the line passing through (,7) and (-1,1). Let (x 1, y 1 ) = (, 7) and (x, y ) = ( 1, 1). The slope is m = y y 1 x x 1 =
More informationAlgebra II. Slide 1 / 261. Slide 2 / 261. Slide 3 / 261. Linear, Exponential and Logarithmic Functions. Table of Contents
Slide 1 / 261 Algebra II Slide 2 / 261 Linear, Exponential and 2015-04-21 www.njctl.org Table of Contents click on the topic to go to that section Slide 3 / 261 Linear Functions Exponential Functions Properties
More informationExponential and logarithmic functions
Exponential and logarithmic functions, حصه الرومي الطالبات.. ندى الغميز, الزهراني مالك المسعد خديجه, األستاذة.. يسرى Modeling with Exponential and Logarithmic Functions Exponential Growth and Decay Models
More informationAlgebra 2 Honors: Final Exam Review
Name: Class: Date: Algebra 2 Honors: Final Exam Review Directions: You may write on this review packet. Remember that this packet is similar to the questions that you will have on your final exam. Attempt
More informationInternet Mat117 Formulas and Concepts. d(a, B) = (x 2 x 1 ) 2 + (y 2 y 1 ) 2., y 1 + y 2. ( x 1 + x 2 2
Internet Mat117 Formulas and Concepts 1. The distance between the points A(x 1, y 1 ) and B(x 2, y 2 ) in the plane is d(a, B) = (x 2 x 1 ) 2 + (y 2 y 1 ) 2. 2. The midpoint of the line segment from A(x
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 informationMath Academy I Fall Study Guide. CHAPTER ONE: FUNDAMENTALS Due Thursday, December 8
Name: Math Academy I Fall Study Guide CHAPTER ONE: FUNDAMENTALS Due Thursday, December 8 1-A Terminology natural integer rational real complex irrational imaginary term expression argument monomial degree
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 informationExponential and Logarithmic Functions. Copyright Cengage Learning. All rights reserved.
Exponential and Logarithmic Functions Copyright Cengage Learning. All rights reserved. 4.5 Exponential and Logarithmic Equations Copyright Cengage Learning. All rights reserved. Objectives Exponential
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 informationExponential and logarithm functions
ucsc supplementary notes ams/econ 11a Exponential and logarithm functions c 2010 Yonatan Katznelson The material in this supplement is assumed to be mostly review material. If you have never studied exponential
More informationFinal Review. Non-calculator problems are indicated. 1. (No calculator) Graph the function: y = x 3 + 2
Algebra II Final Review Name Non-calculator problems are indicated. 1. (No calculator) Graph the function: y = x 3 + 2 2. (No calculator) Given the function y = -2 x + 3-1 and the value x = -5, find the
More information