7.1 Exponential Functions

Size: px
Start display at page:

Download "7.1 Exponential Functions"

Transcription

1 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 it by choosing rational numbers p q that are close to b and evaluating ap/q. We can find rational numbers p q as close to b as we want. As we let our fraction p q approach b, the numbers a p q also approaching a number, we call this number a b. Definition. The function f(x) = a x, with a > 0 and x any real number, is called the exponential function with base a. 2. Use the points to graph the functions f(x) = 2 x and g(x) = ( 1 2 )x. are 1

2 Theorem. Consider the exponential function, f(x) = a x with a > If a > 1, then the graph of a x looks like the graph of 2 x. The higher the value of a the steeper the graph will be. 2. If 0 < a < 1, then the graph of a x looks like the graph of ( 1 2 )x. The smaller the value of a, the steeper the graph will be. 3. If a = 1, then the graph of a x is the horizontal line y = 1. Properties for f(x) = a x : You need to know these properties! 1. The domain of f(x) = a x is (, ). 2. The range of f(x) = a x, if a 1, is (0, ). If a = 1, then the range is the set consisting of the number 1 only. Note: a x can never equal If a > 1, then as x, f(x) = a x, and as x, f(x) = a x If 0 < a < 1, then as x, f(x) = a x 0, and as x, f(x) = a x. 5. If 1 < a and x < y, then a x < a y. Therefore, f(x) = a x is an increasing function if a If 0 < a < 1 and x < y, then a x > a y. Therefore, f(x) = a x is an decreasing function if 0 < a < If a = 1, then f(x) = 1 x = 1, is a constant function. 2

3 3. Which number is bigger: ( 1 3 )6 or ( 1 3 )5? 4. Graph y = 2 x. Does this look graph look familiar? 5. Graph y = 2 2 x. How does this compare to the graph of y = 2 x? 3

4 Consider the graphs below of y = 1.5 x, y = 2 x, y = 2.5 x, y = 3 x, and y = 3.5 x. Also, compare the graphs of y = 0.25 x, y = 0.5 x, and y = 0.75 x. Theorem. In general, we have the following: If a < b, and x > 0, then a x < b x. If a < b, and x < 0, then a x > b x. 6. Which number is larger: (a) 2 2 or 3 2 (b) ( 2 3 )π or ( 3 7 )π (c) ( 5 3 ) 7 or ( ) 7 4

5 The Natural Exponential Function Definition. The natural exponential function is the function f(x) = e x, where e is the limit of (1 + 1 n )n as n goes to infinity. Note. The value of e can be approximated by substituting larger and larger numbers for n in the expression (1 + 1 n )n. A closer approximation is e Then 2 < e < 3 so the graph of e x is between the graphs of 2 x and 3 x. 7. Place the following numbers in increasing order: 2.8 5, 2.3 3, e 4, 35, e 5, Theorem. To evaluate e x, we can use the following limit e x = lim n (1 + x n )n 8. Use the formula e x = lim n (1 + x n )n to approximate e 3 (You need a calculator). n (1 + x n )n 5

6 Example 11. For the following functions, find the domain, range, x-intercepts, and y-intercepts. Also, describe the end behavior of the functions and the intervals whether the exponential is increasing or decreasing. (a) f(x) = 3e x 1 2 Domain: x-intercept(s): y-intercept: increasing or decreasing: Horizontal Asymptotes: (b) g(x) = 2( 1 3 )x+1 18 Domain: x-intercept(s): y-intercept: increasing or decreasing: Horizontal Asymptotes: (c) h(x) = 2(4) x Domain: x-intercept(s): y-intercept: increasing or decreasing: Horizontal Asymptotes: 6

7 7.2 Logarithmic Functions Definition. Let a be any positive number not equal to 1. The logarithm of x to the base a is y if and only if a y = x. The number y is denoted by y = log a x In other words, we say that y is the log of x to the base a, written y = log a x, if when we raise a to the yth power we get x. Note. The logarithm to a base a is the inverse function of a x. The expression log a x is read the log of x to the base a or the log to the base a of x. Remember, to evaluate y = log a x we can write it as an exponential: y = log a x if and only if a y = x 1. Evaluate the following logarithms by writing an exponential equation. (a) log = because (b) log = because (c) log = because (d) log = because (e) log = because (f) log 5 25 = because (g)log 8 64 = because (h) log 4 64 = because (i) log 2 64 = because 2. If the logarithm of x to the base 3 is 4, then x must equal what? 3. Solve the equation log 9 x = 3 for x. 4. If log a 36 = 2, then what is a? 7

8 Theorem. The graph of any logarithmic function y = log a x has the following properties: The fucntion y = log a x has domain (0, ) and the range is (, ). Since a 0 = 1, then log a 1 = 0 and (1, 0) is on the graph. Since a 1 = a, then log a a = 1 and (a, 1) is on the graph. Since a 1 = 1 a, then log a 1 a = 1 and 1 a, 1 is on the graph. The function has a vertical asymptote at x = 0 and no horizontal asymptote. For a > 1, the plot of the logarithmic function with base a, written y = log a x is increasing on its entire domain (0, ) and looks like the following: For 0 < a < 1, the plot of the logarithmic function with base a, written y = log a x is decreasing on its entire domain (0, ) and looks like the following: 5. Find the domain of the function f(x) = e 3x 2 log 3 (2x 5). 6. Graph the following functions. Give the domain, range, and intercepts of the function. (a) f(x) = log 7 x (b) g(x) = 2 log 3 (x 3) 3 (c) h(x) = 3 log 1 (x + 1) 3 8

9 Properties of Logarithms Theorem. Logarithms have the following properties (you need to memorize these): 1. The domain of log a x is (0, ) and its range is (, ). 2. a log a x = x. 3. log a (a x ) = x 4. log a x = log a y if and only if x = y 5. log a 1 = 0 6. log a (xy) = log a x + log a y 7. log a (x y ) = y log a x 8. log a x y = log a x log a y 7. Simplify the expressions: (a) 3 2 log 9 5 = (b) 7 4 log 7 3 = (c) log 4 7 3log log 4 5 (b) 2 log log 6 3 log 6 75 Note. When solving equations with logarithms, we MUST CHECK that our solutions are in the domain of the original equation. 8. Solve the following expressions for x. (a) 3 + log log 5 x = 10 (b) 7 2x+3 = 11 9

10 Natural Logarithm Definition. The natural log function is the log function with base e. Instead of writing log e x, we write ln x. The natural log function ln x is the inverse of the natural exponential function e x. Therefore, ln e x = x and e ln x = x 9. Graph e x and ln x. Give the domain, range, and intercepts of both functions. 10. Suppose that f(x) = ae kx for some value of k and a. Suppose that f(1) = 2 and f(3) = 1. Find a and k. Change of Base Theorem. For any a, b > 0, we can rewrite the logarithmic function log a x using the logarithmic function with base b by the formula log a x = log b x log b a Also, the exponential function a x can be rewritten for base b as a x = b x log b a Note. When we change base, we usually change to base e. So we can rewrite a logarithmic function and exponential function as log a b = ln a ln b and ax = e x ln a Note. Most calculators only have a button for log 10 x and ln x. To evaluate a logarithmic function with any other base, we must use the change of base formula. 11. Use a calculator, to evaluate log Suppose that ln 2 = a, ln 3 = b, ln 5 = c, ln 7 = d, and ln 11 = f. Evaluate and fully simplify the following, writing your answer in terms of a, b, c, d, f. (a) 11 log 6 75 (b) 5 log

11 7.3 Exponential and Logarithmic Equations Solving Exponential Equations: To solve an exponentional equation, use the following steps: 1. Isolate the expression containing the exponent on one side of the equation. 2. Take the logarithm of both sides to bring down the exponent. 3. Solve for the variable. 4. Be careful: we cannot take a logarithm of 0 or a negative number! Example 1. Solve the following equations. (a) x = 7 (b) 5e 3x ( 1 2 )x 3e 3x = 0 (c) 8 e2x 5e 2x + 2 = 3 (d) 3e 2x 14e x + 11 = 0 (e) x 2 2 x x 2 9(2 x ) + 9 = 0. 11

12 Solving Logarithmic Equations: To solve a logarithmic equation, perform the following steps: 1. Isolate the expression containing the logarithm on one side of the equation. 2. Exponentiate both sides using the same base as the log to remove the log function. 3. Solve for the variable. 4. Check: You must check that the solutions are in the domains of your original logarithms! Example 2. Solve the following equations for x. (a) log x = 7 (b) 7 ln(2x 3) = 8 (c) 2 + log 3 (x 1) = log 3 (x 4) (d) log log 5 x = log log 5 (x + 3) (e) 2 ln x = ln 2 + ln(x + 12) 12

13 7.4 Applications of Exponentials and Logarithms Exponential Functions and Population Models There are many species of plants and animals whose populations follow an exponential growth law or exponential growth model. A population of some species satisfies an exponential growth law if there are numbers a and k such that P (t) = P (0)a kt where P (0) represents the population at time t = 0. Note. Notice that P (t) = P (0)a kt can be rewritten as P (t) = P (0)(a k ) t P(t) = P(0) In practice, the separate values of a and k are not important. What is crucial is the value of a k and P (0) (the initial population). If we know both of these, we can compute P (t). Every exponential growth law formula can be expressed in terms of the natural exponential function using the change of base formula: P (t) = P (0)a kt kt ln a = P (0)e 1. If P (t) = 5 4 3t, then P (t) satisfies an exponential growth law. What is P (0)? Also, find a value of t such that P (t) = Suppose that a bacterial colony with an initial population P triples its population every 2 hours. What is the exponential growth model where t is the time in hours? 13

14 3. Which of the following functions satisfy an exponential growth law? (May choose more than one) If it is, find P (0), a, and k. 3t 5 2t 2 3 t t 5 9 t 2 4t t 4. Suppose P (t) satisfies an exponential growth model. If P (5) P (2) = 7, what is a k? If P (0) = 5, determine P (3). Theorem. For a population following an exponential growth law P (t) = P (0)a kt, if we know the population at two times b and c, then we can calculate a k. Notice P (b) P (c) = P (0)akb P (0)a kc = akb kc = (a k ) (b c). So a k = ( P (b) P (a) ) 1 b c. If we also know P(0), then we can calculate P (t) for any t. 5. A biologist decides that an epidemic spreads through a population of a city according to the following model p(t) = 1 e 0.34t, where p(t) represents that fraction of the citys population which has come down with the disease, and t is in weeks. How long will it take for 90% of the city to become infected? 14

15 Exponential Functions and Radioactive Decay There are many material substances which decay radioactively. That is, they spontaneously change into a different material, and in the decay process emit charged particles. The rate of decay is measured as the half-life of the substance. Definition. The half-life of a substance is how long it takes for half of the substance to decay. Theorem. If an element decays radioactively, then the amount of this element at any time t satisfies an exponential growth/decay law. That is, if A(t) denotes the amount of material at time t, then A(t) = A(0)e kt. Since this is a decay, then the function is decreasing (so k < 0 and e k < 1). 6. Using the fact that the half-life of carbon 14 is 5800 years, determine the exponential growth/decay model of 14 C. 7. If the half-life of a substance is 5 years, how many years will it take for 2 pounds of this substance to decay to 1 8 of a pound? 15

16 8. The half-life of uranium 235 is years. If we start out with 1.5 kilograms of 235U in 2010, how much uranium will be left after 10,000 years. 9. Suppose a radioactive substance satisfies the exponential growth/decay law A(t) = A(0)7 3t, where t is in decades. What is the half-life of this substance? 10. What is the half-life of a sample that decayed 23% after 4 years? Do not forget your units. 16

17 Interest Compounded Continuously Theorem. If an initial principal of P is invested at an annual rate of r and interest is compounded continuously, then the amount A(t) in the account after t years is given by A(t) = P e rt Suppose an investment is earning 9% compounded continuously. How long will it take for the investment to double? 12. Suppose that an investment is compounded continuously at a constant annual rate, and grows from $1000 to $1500 over 3 years. How long would it take $5000 to grow to $14,000 if it was invested at the same rate? 13. An object is placed in a refrigerator, and the temperature of an object T (in C) after t minutes is given by T (t) = e kt. After 15 minutes, the temperature of the object is 10 C. (a) Find the value of k. (b) What was the starting temperature of the object, and how long will it take the object to reach a temperature of 5 C? 17

Chapter 6: Exponential and Logarithmic Functions

Chapter 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 information

Skill 6 Exponential and Logarithmic Functions

Skill 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

Notes for exponential functions The week of March 6. Math 140

Notes for exponential functions The week of March 6. Math 140 Notes for exponential functions The week of March 6 Math 140 Exponential functions: formulas An exponential function has the formula f (t) = ab t, where b is a positive constant; a is the initial value

More information

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.

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. 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 information

Section Exponential Functions

Section 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 information

Section 4.2 Logarithmic Functions & Applications

Section 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 information

Skill 6 Exponential and Logarithmic Functions

Skill 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

The Exponential function f with base b is f (x) = b x where b > 0, b 1, x a real number

The 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 information

Pre-Calculus Final Exam Review Units 1-3

Pre-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 information

2.6 Logarithmic Functions. Inverse Functions. Question: What is the relationship between f(x) = x 2 and g(x) = x?

2.6 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 information

HW#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. 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 information

Exponential and Logarithmic Functions

Exponential 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 information

Inverse 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 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 information

5.1. EXPONENTIAL FUNCTIONS AND THEIR GRAPHS

5.1. 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 information

Name: Partners: PreCalculus. Review 5 Version A

Name: Partners: PreCalculus. Review 5 Version A Name: Partners: PreCalculus Date: Review 5 Version A [A] Circle whether each statement is true or false. 1. 3 log 3 5x = 5x 2. log 2 16 x+3 = 4x + 3 3. ln x 6 + ln x 5 = ln x 30 4. If ln x = 4, then e

More information

The above statement is the false product rule! The correct product rule gives g (x) = 3x 4 cos x+ 12x 3 sin x. for all angles θ.

The above statement is the false product rule! The correct product rule gives g (x) = 3x 4 cos x+ 12x 3 sin x. for all angles θ. Math 7A Practice Midterm III Solutions Ch. 6-8 (Ebersole,.7-.4 (Stewart DISCLAIMER. This collection of practice problems is not guaranteed to be identical, in length or content, to the actual exam. You

More information

4 Exponential and Logarithmic Functions

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 information

PRECAL REVIEW DAY 11/14/17

PRECAL 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 information

Exponential Growth (Doubling Time)

Exponential Growth (Doubling Time) Exponential Growth (Doubling Time) 4 Exponential Growth (Doubling Time) Suppose we start with a single bacterium, which divides every hour. After one hour we have 2 bacteria, after two hours we have 2

More information

10 Exponential and Logarithmic Functions

10 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 information

GUIDED NOTES 6.1 EXPONENTIAL FUNCTIONS

GUIDED 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 information

Practice Questions for Final Exam - Math 1060Q - Fall 2014

Practice 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 information

Logarithms involve the study of exponents so is it vital to know all the exponent laws.

Logarithms involve the study of exponents so is it vital to know all the exponent laws. Pre-Calculus Mathematics 12 4.1 Exponents Part 1 Goal: 1. Simplify and solve exponential expressions and equations Logarithms involve the study of exponents so is it vital to know all the exponent laws.

More information

for every x in the gomain of g

for 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 information

Review questions for Math 111 final. Please SHOW your WORK to receive full credit Final Test is based on 150 points

Review questions for Math 111 final. Please SHOW your WORK to receive full credit Final Test is based on 150 points Please SHOW your WORK to receive full credit Final Test is based on 150 points 1. True or False questions (17 pts) a. Common Logarithmic functions cross the y axis at (0,1) b. A square matrix has as many

More information

Chapter 2 Functions and Graphs

Chapter 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 information

CHAPTER FIVE. Solutions for Section 5.1. Skill Refresher. Exercises

CHAPTER 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 information

Exponential and Logarithmic Functions

Exponential 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 information

Composition of Functions

Composition 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 information

Exponential and Logarithmic Functions. 3. Pg #17-57 column; column and (need graph paper)

Exponential 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 information

Chapter 11 Logarithms

Chapter 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 information

Exponential functions are defined and for all real numbers.

Exponential functions are defined and for all real numbers. 3.1 Exponential and Logistic Functions Objective SWBAT evaluate exponential expression and identify and graph exponential and logistic functions. Exponential Function Let a and b be real number constants..

More information

in terms of p, q and r.

in terms of p, q and r. Logarithms and Exponents 1. Let ln a = p, ln b = q. Write the following expressions in terms of p and q. ln a 3 b ln a b 2. Let log 10 P = x, log 10 Q = y and log 10 R = z. Express P log 10 QR 3 2 in terms

More information

UNIT 5: DERIVATIVES OF EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS. Qu: What do you remember about exponential and logarithmic functions?

UNIT 5: DERIVATIVES OF EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS. Qu: What do you remember about exponential and logarithmic functions? UNIT 5: DERIVATIVES OF EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS 5.1 DERIVATIVES OF EXPONENTIAL FUNCTIONS, y = e X Qu: What do you remember about exponential and logarithmic functions? e, called Euler s

More information

GUIDED NOTES 6.1 EXPONENTIAL FUNCTIONS

GUIDED 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 information

April 9, 2009 Name The problems count as marked. The total number of points available is 160. Throughout this test, show your work.

April 9, 2009 Name The problems count as marked. The total number of points available is 160. Throughout this test, show your work. April 9, 009 Name The problems count as marked The total number of points available is 160 Throughout this test, show your work 1 (15 points) Consider the cubic curve f(x) = x 3 + 3x 36x + 17 (a) Build

More information

Exponential and. Logarithmic Functions. Exponential Functions. Logarithmic Functions

Exponential and. Logarithmic Functions. Exponential Functions. Logarithmic Functions Chapter Five Exponential and Logarithmic Functions Exponential Functions Logarithmic Functions Properties of Logarithms Exponential Equations Exponential Situations Logarithmic Equations Exponential Functions

More information

SHORT ANSWER. Answer the question, including units in your answer if needed. Show work and circle your final answer.

SHORT ANSWER. Answer the question, including units in your answer if needed. Show work and circle your final answer. Math 131 Group Review Assignment (5.5, 5.6) Print Name SHORT ANSWER. Answer the question, including units in your answer if needed. Show work and circle your final answer. Solve the logarithmic equation.

More information

Objectives. 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 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 y

x y (a) The curve y = ax n, where a and n are constants, passes through the points (2.25, 27), (4, 64) and (6.25, p). Calculate the value of a, of n and of p. [5] (b) The mass, m grams, of a radioactive substance

More information

Section 6.1: Composite Functions

Section 6.1: Composite Functions Section 6.1: Composite Functions Def: Given two function f and g, the composite function, which we denote by f g and read as f composed with g, is defined by (f g)(x) = f(g(x)). In other words, the function

More information

Exponential and Logarithmic Functions. Copyright Cengage Learning. All rights reserved.

Exponential 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 information

Section 5.1 Determine if a function is a polynomial function. State the degree of a polynomial function.

Section 5.1 Determine if a function is a polynomial function. State the degree of a polynomial function. Test Instructions Objectives Section 5.1 Section 5.1 Determine if a function is a polynomial function. State the degree of a polynomial function. Form a polynomial whose zeros and degree are given. Graph

More information

MA123, Supplement: Exponential and logarithmic functions (pp , Gootman)

MA123, Supplement: Exponential and logarithmic functions (pp , Gootman) MA23, Supplement: Exponential an logarithmic functions pp. 35-39, Gootman) Chapter Goals: Review properties of exponential an logarithmic functions. Learn how to ifferentiate exponential an logarithmic

More information

Logarithmic Functions and Models Power Functions Logistic Function. Mathematics. Rosella Castellano. Rome, University of Tor Vergata

Logarithmic 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 information

2. (12 points) Find an equation for the line tangent to the graph of f(x) =

2. (12 points) Find an equation for the line tangent to the graph of f(x) = November 23, 2010 Name The total number of points available is 153 Throughout this test, show your work Throughout this test, you are expected to use calculus to solve problems Graphing calculator solutions

More information

MATH 1431-Precalculus I

MATH 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 information

Honors 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 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 information

CC2 Exponential v.s. Log Functions

CC2 Exponential v.s. Log Functions CC2 Exponential v.s. Log Functions CC1 Mastery Check Error Analysis tomorrow Retake? TBA (most likely end of this week) *In order to earn the chance for re-assessment, you must complete: Error Analysis

More information

Example. 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) = 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 information

INTERNET 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. (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 information

17 Exponential and Logarithmic Functions

17 Exponential and Logarithmic Functions 17 Exponential and Logarithmic Functions Concepts: Exponential Functions Power Functions vs. Exponential Functions The Definition of an Exponential Function Graphing Exponential Functions Exponential Growth

More information

Population Changes at a Constant Percentage Rate r Each Time Period

Population Changes at a Constant Percentage Rate r Each Time Period Concepts: population models, constructing exponential population growth models from data, instantaneous exponential growth rate models, logistic growth rate models. Population can mean anything from bacteria

More information

notes.notebook April 08, 2014

notes.notebook April 08, 2014 Chapter 7: Exponential Functions graphs solving equations word problems Graphs (Section 7.1 & 7.2): c is the common ratio (can not be 0,1 or a negative) if c > 1, growth curve (graph will be increasing)

More information

nt and A = Pe rt to solve. 3) Find the accumulated value of an investment of $10,000 at 4% compounded semiannually for 5 years.

nt 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 information

THE EXPONENTIAL AND NATURAL LOGARITHMIC FUNCTIONS: e x, ln x

THE EXPONENTIAL AND NATURAL LOGARITHMIC FUNCTIONS: e x, ln x Mathematics Revision Guides The Exponential and Natural Log Functions Page 1 of 17 M.K. HOME TUITION Mathematics Revision Guides Level: A-Level Year 1 / AS THE EXPONENTIAL AND NATURAL LOGARITHMIC FUNCTIONS:

More information

Fundamentals of Mathematics (MATH 1510)

Fundamentals of Mathematics (MATH 1510) Fundamentals of Mathematics () Instructor: Email: shenlili@yorku.ca Department of Mathematics and Statistics York University February 3-5, 2016 Outline 1 growth (doubling time) Suppose a single bacterium

More information

Concept Category 2. Exponential and Log Functions

Concept 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 information

Graphing Exponentials 6.0 Topic: Graphing Growth and Decay Functions

Graphing 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 information

Part 4: Exponential and Logarithmic Functions

Part 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 information

3. Solve the following inequalities and express your answer in interval notation.

3. Solve the following inequalities and express your answer in interval notation. Youngstown State University College Algebra Final Exam Review (Math 50). Find all Real solutions for the following: a) x 2 + 5x = 6 b) 9 x2 x 8 = 0 c) (x 2) 2 = 6 d) 4x = 8 x 2 e) x 2 + 4x = 5 f) 36x 3

More information

Exponential Functions Dr. Laura J. Pyzdrowski

Exponential Functions Dr. Laura J. Pyzdrowski 1 Names: (4 communication points) About this Laboratory An exponential function is an example of a function that is not an algebraic combination of polynomials. Such functions are called trancendental

More information

Logarithmic Functions and Their Graphs

Logarithmic Functions and Their Graphs Section 3. Logarithmic Functions and Their Graphs Look at the graph of f(x) = x Does this graph pass the Horizontal Line Test? es What does this mean? that its inverse is a function Find the inverse of

More information

INTERNET 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 + 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 information

Homework 3. (33-40) The graph of an exponential function is given. Match each graph to one of the following functions.

Homework 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 information

Mock 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) 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 information

Intermediate Algebra Chapter 12 Review

Intermediate 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 information

7.1. Calculus of inverse functions. Text Section 7.1 Exercise:

7.1. Calculus of inverse functions. Text Section 7.1 Exercise: Contents 7. Inverse functions 1 7.1. Calculus of inverse functions 2 7.2. Derivatives of exponential function 4 7.3. Logarithmic function 6 7.4. Derivatives of logarithmic functions 7 7.5. Exponential

More information

8-1 Exploring Exponential Models

8-1 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 information

Logarithmic, Exponential, and Other Transcendental Functions. Copyright Cengage Learning. All rights reserved.

Logarithmic, 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 information

Solutions to MAT 117 Test #3

Solutions to MAT 117 Test #3 Solutions to MAT 7 Test #3 Because there are two versions of the test, solutions will only be given for Form C. Differences from the Form D version will be given. (The values for Form C appear above those

More information

Name Advanced Math Functions & Statistics. Non- Graphing Calculator Section A. B. C.

Name Advanced Math Functions & Statistics. Non- Graphing Calculator Section A. B. C. 1. Compare and contrast the following graphs. Non- Graphing Calculator Section A. B. C. 2. For R, S, and T as defined below, which of the following products is undefined? A. RT B. TR C. TS D. ST E. RS

More information

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

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 information

Math 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 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 information

Evaluate the expression using the values given in the table. 1) (f g)(6) x f(x) x g(x)

Evaluate 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 information

Math Analysis - Chapter 5.4, 5.5, 5.6. (due the next day) 5.4 Properties of Logarithms P.413: 7,9,13,15,17,19,21,23,25,27,31,33,37,41,43,45

Math Analysis - Chapter 5.4, 5.5, 5.6. (due the next day) 5.4 Properties of Logarithms P.413: 7,9,13,15,17,19,21,23,25,27,31,33,37,41,43,45 Math Analysis - Chapter 5.4, 5.5, 5.6 Mathlete: Date Assigned Section Homework (due the next day) Mon 4/17 Tue 4/18 5.4 Properties of Logarithms P.413: 7,9,13,15,17,19,21,23,25,27,31,33,37,41,43,45 5.5

More information

2. Algebraic functions, power functions, exponential functions, trig functions

2. Algebraic functions, power functions, exponential functions, trig functions Math, Prep: Familiar Functions (.,.,.5, Appendix D) Name: Names of collaborators: Main Points to Review:. Functions, models, graphs, tables, domain and range. Algebraic functions, power functions, exponential

More information

Sec. 4.2 Logarithmic Functions

Sec. 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

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. B) 6x + 4

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. B) 6x + 4 Math1420 Review Comprehesive Final Assessment Test Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Add or subtract as indicated. x + 5 1) x2

More information

Review of Functions A relation is a function if each input has exactly output. The graph of a function passes the vertical line test.

Review 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 information

1.3 Exponential Functions

1.3 Exponential Functions 22 Chapter 1 Prerequisites for Calculus 1.3 Exponential Functions What you will learn about... Exponential Growth Exponential Decay Applications The Number e and why... Exponential functions model many

More information

SBS Chapter 2: Limits & continuity

SBS Chapter 2: Limits & continuity SBS Chapter 2: Limits & continuity (SBS 2.1) Limit of a function Consider a free falling body with no air resistance. Falls approximately s(t) = 16t 2 feet in t seconds. We already know how to nd the average

More information

GOOD 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.

GOOD 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 information

Chapter 3. Exponential and Logarithmic Functions. 3.2 Logarithmic Functions

Chapter 3. Exponential and Logarithmic Functions. 3.2 Logarithmic Functions Chapter 3 Exponential and Logarithmic Functions 3.2 Logarithmic Functions 1/23 Chapter 3 Exponential and Logarithmic Functions 3.2 4, 8, 14, 16, 18, 20, 22, 30, 31, 32, 33, 34, 39, 42, 54, 56, 62, 68,

More information

Logarithmic Functions

Logarithmic 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 information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Calculus I - Homework Chapter 2 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether the graph is the graph of a function. 1) 1)

More information

Chapter 3 Exponential and Logarithmic Functions

Chapter 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 information

Algebra II. Slide 1 / 261. Slide 2 / 261. Slide 3 / 261. Linear, Exponential and Logarithmic Functions. Table of Contents

Algebra 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 information

MAC Module 8. Exponential and Logarithmic Functions I. Learning Objectives. - Exponential Functions - Logarithmic Functions

MAC 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 information

MAC Module 8 Exponential and Logarithmic Functions I. Rev.S08

MAC 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 information

Teacher: Mr. Chafayay. Name: Class & Block : Date: ID: A. 3 Which function is represented by the graph?

Teacher: 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 information

The 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.

The 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 information

4.1 Exponential Functions

4.1 Exponential Functions Graduate T.A. Department of Mathematics Dynamical Systems and Chaos San Diego State University April 9, 211 Definitions The functions that involve some combinations of basic arithmetic operations, powers,

More information

Chapter 3 Exponential and Logarithmic Functions

Chapter 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 information

Exponential and Logarithmic Functions

Exponential and Logarithmic Functions Öğr. Gör. Volkan ÖĞER FBA 1021 Calculus 1/ 40 Exponential and Logarithmic Functions Exponential Functions The functions of the form f(x) = b x, for constant b, are important in mathematics, business, economics,

More information

Unit 5: Exponential and Logarithmic Functions

Unit 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 information

p351 Section 5.5: Bases Other than e and Applications

p351 Section 5.5: Bases Other than e and Applications p351 Section 5.5: Bases Other than e and Applications Definition of Exponential Function to Base a If a is a positive real number (a 1) and x is any real number, then the exponential function to the base

More information

Transformations of Functions and Exponential Functions January 24, / 35

Transformations 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

Math 137 Exam #3 Review Guide

Math 137 Exam #3 Review Guide Math 7 Exam # Review Guide The third exam will cover Sections.-.6, 4.-4.7. The problems on this review guide are representative of the type of problems worked on homework and during class time. Do not

More information

NONLINEAR FUNCTIONS A. Absolute Value Exercises: 2. We need to scale the graph of Qx ( )

NONLINEAR FUNCTIONS A. Absolute Value Exercises: 2. We need to scale the graph of Qx ( ) NONLINEAR FUNCTIONS A. Absolute Value Eercises:. We need to scale the graph of Q ( ) f ( ) =. The graph is given below. = by the factor of to get the graph of 9 - - - - -. We need to scale the graph of

More information