Another enormous super-family of functions are exponential functions.

Size: px
Start display at page:

Download "Another enormous super-family of functions are exponential functions."

Transcription

1 Hartfield College Algebra (Version Thomas Hartfield) Unit FIVE Page of 39 Topic 37: Exponential Functions In previous topics we ve discussed power functions, n functions of the form f x x, where n is a nonzero real number. Squaring and cubing functions, along with all of the even or odd powered functions, are obviously power functions, but so are root functions (n is a rational number for the power) and reciprocal functions (n is a negative integer). Another enormous super-family of functions are exponential functions. Definition: As exponential function with base b f x b x b b is defined as, 0, 1. Domain:, Range: 0,

2 Hartfield College Algebra (Version Thomas Hartfield) Unit FIVE Page of 39 Examples of Basic Exponential Functions Observations about exponential functions: 1. f(0) = 1. All basic exponential functions have a y-intercept at (0, 1). 2. All basic exponential functions have a horizontal asymptote of y = 0. f x 2 x f x 5 x 3a. When b > 1, a basic exponential function is always increasing. The right end goes up (as x the function increases without bound) and the left end asymptotically approaches the x-axis (as x the function approaches 0). 3b. When 0 < b < 1, a basic exponential function is always decreasing. The left end goes up (as x the function increases without bound) and the right end asymptotically approaches the x-axis (as x the function approaches 0). 4. The closer b is to 1, the more linear the graph appears. f x 0.5 x f x 0.8 x

3 Hartfield College Algebra (Version Thomas Hartfield) Unit FIVE Page of 39 Natural Exponential Functions and the Number e When discussing exponential functions, it is important to introduce a special number that shows up frequently in mathematics, the number e. Definition: n 1 e lim n n A very special basic exponential function is one which uses e as its base. Definition: The natural exponential function is an exponential function with base e: x e. f x Now what does this definition mean? A limit (the abbreviation lim) describes the behavior of an expression as a variable (in this case n) changes in a certain way. If n gets larger and larger, without ceasing, the expression here gets closer and closer to the value of e. The function behaves all of the basic rules of exponential functions, including the fact that the domain is, and the range is 0,, and since e > 1, it satisfies observation 3a made previously. We refer to e by several names, including Napier s Constant or Euler s Number, but most frequently it is identified as the Natural Base.

4 Hartfield College Algebra (Version Thomas Hartfield) Unit FIVE Page of 39 Transformations of basic exponential functions The algebraic transformations we covered in a previous unit are possible with exponential functions as well. Ex. 1 Identify the basic function in the given function. Determine the transformations applied to the basic function that produce the given function. State the transformations applied, in the correct order, using units and directions as appropriate. Finally, determine the domain, range, and asymptote of the given function. f( x) 32 x 2 Ex. 2 Identify the basic function in the given function. Determine the transformations applied to the basic function that produce the given function. State the transformations applied, in the correct order, using units and directions as appropriate. Finally, determine the domain, range, and asymptote of the given function. f( x) 63 x 1

5 Hartfield College Algebra (Version Thomas Hartfield) Unit FIVE Page of 39 Ex. 3 Determine the simplest exponential function that satisfies the conditions below. Ex. 4 Determine the simplest exponential function that satisfies the conditions below. a. Has a y-intercept of (0, 1) and passes 1 through the point 1,. 2 a. Has a y-intercept of (0, 4) and passes b. Has a y-intercept of (0, 1) and passes 9 through the point 2,. 4 b. Has a y-intercept of (0, 3) and passes

6 Hartfield College Algebra (Version Thomas Hartfield) Unit FIVE Page of 39 Topic 38: Logarithmic Functions & Expressions Since basic exponential functions are one-to-one, that means they are invertible. This leads us to the definition of another super-family of functions which are the inverses of exponentials. Ex. 1 Using the graph of f( x) 2 x below, sketch the graph of gx ( ) log 2 x. Definition: The inverse of an exponential x function f x b is called a base b logarithmic function and notated as f 1 x log b x (which is read as log base b of x).

7 Hartfield College Algebra (Version Thomas Hartfield) Unit FIVE Page of 39 below, 2 sketch the graph of gx ( ) log 1 x. 2 1 Ex. 2 Using the graph of f( x) x Conclusions that can be drawn about logarithmic functions. 1. Domain is 0, and Range is,. 2. g(1) = 0. All basic logarithmic functions have an x-intercept at (1, 0) and have a vertical asymptote of x = When b > 1, a basic logarithmic function is always increasing and when 0 < b < 1, a basic logarithmic function is always decreasing. Two special types of logarithmic functions are the common logarithm whose base is 10, log 10 x, which is usually written as just log x and the natural logarithm whose base is e, log e x, which is usually written as just ln x.

8 Hartfield College Algebra (Version Thomas Hartfield) Unit FIVE Page of 39 Transformations are also possible with logarithmic functions. Ex. 3 Identify the basic function in the given function. Determine the transformations applied to the basic function that produce the given function. State the transformations applied, in the correct order, using units and directions as appropriate. Finally, determine the domain, range, and asymptote of the given function. f( x) log x 3 2 Ex. 4 Identify the basic function in the given function. Determine the transformations applied to the basic function that produce the given function. State the transformations applied, in the correct order, using units and directions as appropriate. Finally, determine the domain, range, and asymptote of the given function. f( x) 5log x 1 2

9 Hartfield College Algebra (Version Thomas Hartfield) Unit FIVE Page of 39 Let s explore logarithmic expressions (which we can just call logarithms) in depth. An essential property of logarithms is the following: n b c log b( c) n. That is, logb (c) is the exponent you apply to b to make c. Ex. 6 Evaluate each logarithm. a. log6 36 Ex. 5 Evaluate each logarithm. a. log7 1 b. log3 81 b. log1111 c. log25 5

10 Hartfield College Algebra (Version Thomas Hartfield) Unit FIVE Page of 39 Ex. 7 Evaluate each logarithm. Ex. 8 Evaluate each logarithm. a. log9 27 a. log b. log b. log c. log c. log5 0.04

11 Hartfield College Algebra (Version Thomas Hartfield) Unit FIVE Page of 39 Topic 39: Properties of Logarithms In addition to the property of logarithms established previously, there are three additional properties of logarithms (similar to properties of exponentials and sometimes called laws of logarithms ): Product Rule for exponentials: m n m n a a a Product Rule for logarithms: log xy log x log y a a a It is possible to take a single relatively complicated logarithm and rewrite it into a collection of simpler logarithmic expressions or constants by using the properties of logarithms. Ex. 1 Expand the expression using the properties of logarithms. Simplify where possible. log2 4 8x y a mn Quotient Rule for exponentials: a n a Quotient Rule for logarithms: log x log x log y m Power Rule for exponentials: n n Power Rule for logarithms: loga m a a a a m n a x nlog x a y Deconstructing a single logarithm:

12 Hartfield College Algebra (Version Thomas Hartfield) Unit FIVE Page of 39 Ex. 2 Expand the expression using the properties of logarithms. Simplify where possible. Ex. 3 Expand the expression using the properties of logarithms. Simplify where possible. 2 ln a ec 5 log 7 x y z

13 Hartfield College Algebra (Version Thomas Hartfield) Unit FIVE Page of 39 Ex. 4 Expand the expression using the properties of logarithms. Simplify where possible. 9 v log3 9 n Reconstructing a single logarithm: This is the opposite of the last skill. Sometimes a single logarithm may be a preferable expression. Ex. 5 Rewrite as a single logarithm. 3log x4log y 5log z Ex. 6 Rewrite as a single logarithm. log 2 6log ulog v

14 Hartfield College Algebra (Version Thomas Hartfield) Unit FIVE Page of 39 Ex. 7 Rewrite as a single logarithm. Ex. 9 Rewrite as a single logarithm. 3log x log x 2 log x 1 log x log xlog x Ex. 8 Rewrite as a single logarithm. 2log x log x1 log 3 log 2x Ex. 10 Rewrite as a single logarithm. log xlog x 2 3 9

15 Hartfield College Algebra (Version Thomas Hartfield) Unit FIVE Page of 39 Using the properties of logarithms it should be relatively easy to evaluate log2 32. Change of Base Formula log a n log log b b n a Calculators will easy handle approximating log32 and ln32. But how do we determine the approximate value of log 32? 3 At the very least we should be able to determine between which two integers the value should fall. This formula allows us to calculate the approximate value of logarithms where the exact value is difficult or impossible to express as a non-logarithmic expression. Since we can choose what we change the base into, the most logical choices are usually base 10 or base e, thus: log a n ln ln n a log log n a Ex. 11 Approximate the value of 3 log 32, five places after the decimal point.

16 Hartfield College Algebra (Version Thomas Hartfield) Unit FIVE Page of 39 Topic 40: Solving Logarithmic Equations Ex. 1 Solve. A logarithmic equation is an equation involving one or more logarithmic expressions. x x log 5 1 log Logarithmic equations can be divided into two families: 1. Those equations that can be simplified into the form of a logarithmic expression equal to a number. 2. Those equations that can be simplified into the form of two logarithmic equations (same base) equal to each other. To solve the first case, apply the following steps: 1. Use the laws of logarithms, as necessary, to combine logarithms. 2. Rewrite the logarithm equation into exponential form. 3. Solve using any appropriate algebraic and/or arithmetic methods. 4. Check your solutions. Like radical equations, logarithmic equations can (and often do) produce extraneous solutions.

17 Hartfield College Algebra (Version Thomas Hartfield) Unit FIVE Page of 39 Ex. 2 Solve. Ex. 3 Solve. x x log 7 log log 3 x 1 5

18 Hartfield College Algebra (Version Thomas Hartfield) Unit FIVE Page of 39 Ex. 4 Solve. x x log 1 log If the equation is simplified (using the laws of logarithms) in such a way both sides have a logarithmic expression, consider the limited possibilities for making the equation true... Ex. 5 Solve. x x ln 2 ln 3 ln 4

19 Hartfield College Algebra (Version Thomas Hartfield) Unit FIVE Page of 39 Ex. 6 Solve. x x x x log 4 log 5 log 1 log 7

20 Hartfield College Algebra (Version Thomas Hartfield) Unit FIVE Page of 39 Ex. 7 Solve. x x x x log 1 log 2 log log

21 Hartfield College Algebra (Version Thomas Hartfield) Unit FIVE Page of 39 Topic 41: Solving Exponential Equations (Part 1) Ex. 1 Solve. An exponential equation is an equation involving one or more exponential expressions. 3x Basic process to solve most exponential equations: 1. Isolate an exponential expression on one side. 2. Take the natural logarithm (or common log) of both sides. 3. Use the laws of logarithms to rewrite the exponential expression so that no variable remains in the exponent. 4. Apply basic algebraic and arithmetic manipulation to solve for x. 5. Use the laws of logarithms to simplify the solution and approximate the solution. 6. Check your solution. With regards to step 5, I will expect you to further rewrite the solution into a single logarithmic expression and approximate five places after the decimal.

22 Hartfield College Algebra (Version Thomas Hartfield) Unit FIVE Page of 39 Ex. 2 Solve. Ex. 3 Solve. x x

23 Hartfield College Algebra (Version Thomas Hartfield) Unit FIVE Page of 39 Ex. 4 Solve. Ex. 5 Solve. 2x x 5e 3 20

24 Hartfield College Algebra (Version Thomas Hartfield) Unit FIVE Page of 39 Topic 42: Solving Exponential Equations (Part 2) Ex. 2 Solve. If the bases are equal or the bases are both powers of the same constant, solving an equation of two exponential expressions is fairly easy. x2 2x1 8 4 Ex. 1 Solve. x3 2x1 5 5

25 Hartfield College Algebra (Version Thomas Hartfield) Unit FIVE Page of 39 If the bases are neither equal nor related, the process for solving an exponential equation involving two exponential expressions is similar (but more involved) than those covered in a previous topic. Ex. 3 Solve. 2x3 x1 3 2

26 Hartfield College Algebra (Version Thomas Hartfield) Unit FIVE Page of 39 Ex. 4 Solve. x1 2x 4 3

27 Hartfield College Algebra (Version Thomas Hartfield) Unit FIVE Page of 39 Some exponential equations be solved using factoring (with substitution sometimes helpful). The important detail to remember here is 2x x that 2 e e. Ex. 6 Solve. e 2x x 4e 0 Ex. 5 Solve. e 2x x e 20

28 Hartfield College Algebra (Version Thomas Hartfield) Unit FIVE Page of 39 Topic 43: Compound Interest There are two primary formulas used for problems involving compounded interest. Formula 1 Discretely Compounded Interest P1 At r n nt t = units of time in years P = principle (initial value) r = rate of interest as a decimal A(t) = amount at/after time t n = number of compounds per year This is the formula used most often when interest is compounded. In this formula compounding occurs at some defined regular intervals. Examples include quarterly (n = 4), monthly (n = 12), and weekly (n = 52), amongst others. A second formula for compounding interest, which involves the natural base, is used only in cases where interest is compounded continuously. Formula 2 Continuously Compounded Interest At r t Pe t = units of time in years P = principle (initial value) r = rate of interest as a decimal A(t) = amount at/after time t

29 Hartfield College Algebra (Version Thomas Hartfield) Unit FIVE Page of 39 Given information about all constants in the formulas except one, it is possible to solve for that unknown. For example: If given everything but A(t), you can find a future amount from a present amount. If given everything but P, you can find the present amount needed for a future amount. If given everything but t, you can solve an equation to find the time needed to go from a present amount to a future amount. Ex. 1a Jordan wishes to invest $ He is looking at two different Certificates of Deposit (CDs), each of which compounds interest monthly. The two year CD earns 1.20% interest while the five year CD earns 2.25% interest. How much will each CD be worth at maturation? Express to the penny.

30 Hartfield College Algebra (Version Thomas Hartfield) Unit FIVE Page of 39 Ex. 1b If Jordan wants to have $3000 in the future with the five year CD, how much would he have to invest now? Express to the penny. Ex. 1c If Jordan is offered a savings account earning 1.5% interest compounded quarterly, how long would it take a $2500 investment to double? Express to the nearest tenth of a year.

31 Hartfield College Algebra (Version Thomas Hartfield) Unit FIVE Page of 39 Ex. 2a A furniture company offers a promotion where no interest is accrued on an account for four years. In the fine print you find that if the account is not paid in full within the four year period, all of the interest that could have accumulated on the original amount will retroactively be applied to the account. A $3 000 sofa is not paid off in time. How much interest would be tacked on to the account if the interest is 15.99% compounded biweekly? Express to the dollar. A term often used in finance, associated with lending and savings, is Annual Percentage Rate, or APR. The APR of an account is a measure of what an equivalent rate would be if the interest was only compounded annually (this provided a clearer picture on the long term effect of the nominal rate.) To calculate the APR on an account, let P = $1 and t = 1 year. Then subtract the original dollar from A(1) and convert to a percentage. Ex. 2b What is the APR on the furniture company s promotional offer? Express to the nearest hundredth of a percent.

32 Hartfield College Algebra (Version Thomas Hartfield) Unit FIVE Page of 39 Ex. 3a Ashley has invested $4000 in an interest-bearing savings account which earns 3.75% interest compounded quarterly. How much will she have in 5 years? Express to the penny. Ex. 3b How long will it take for Ashley s investment to be worth $10,000? Express to the tenth of a year.

33 Hartfield College Algebra (Version Thomas Hartfield) Unit FIVE Page of 39 Ex. 4 Taylor s grandmother put away money for him in a municipal bond on the day he was born, with instructions that he should only cash it in when it had tripled in value. If the bond earns 5.2% interest, compounded weekly, how old will he be when he can cash it in according to her wishes? Ex. 5 Shannon wants her investment of $10,000 to be valued at $30,000 in 10 years. What interest rate, expressed to the hundredth of a percent, will accomplish this if the interest is compounded continuously?

34 Hartfield College Algebra (Version Thomas Hartfield) Unit FIVE Page of 39 Topic 44: Other Applications of Exponentials Radioactive Decay A variety of application problems can use formulas with exponentials in them. Population Growth based on a continuous rate The growth (or decline of a population) can be r t determined by the formula n t n 0 e. n 0 = initial population n(t) = population at time t t =time (any units) r = relative rate of growth per unit of time The amount of radioactive material in a sample can be determined by the formula 0 2 mt m t h. m 0 = initial amount of radioactive material m(t) = amount of radioactive material at time t t = time h = half-life of radioactive material (units for time and half-life must agree) Newton s Law of Cooling The temperature of an item after being in an environment of a different temperature can be kt found by the formula T t AT Ae 0. T 0 = initial item temp. T(t) = temperature at time t A = temperature of the environment t = time (any units) k = cooling rate

35 Hartfield College Algebra (Version Thomas Hartfield) Unit FIVE Page of 39 Ex. 1 An abandoned island, overrun with rats, is used as a dumping ground. The rat population is estimated by the function 0.15t n t 45 e, where t is the number of years after 2005 & n(t) is in thousands. a: According to the formula, what was the population of rats in 2005? Ex. 2 According to projections by the state of Georgia, Forsyth County had about population of 183 thousand in 2010 and is expected to have a population of about 264 thousand in What is the expected relative rate of growth in Forsyth County over the decade from 2010 to 2020? Express to the tenth of a percent. b: According to the formula, what was the relative growth rate in the rat population? c: Estimate the number of rats on the island in Express to the tenth of a thousand.

36 Hartfield College Algebra (Version Thomas Hartfield) Unit FIVE Page of 39 Ex. 3 Morgan wishes to do an experiment with fruit flies. At 9 am he has 200 flies and under the controls of his experiment their population should increase at a relative rate of 35% per hour. a: Estimate the population of fruit flies at 4 pm of the same day. Express to the whole fly. b: At what time, to the nearest minute, will his population reach 5000 flies?

37 Hartfield College Algebra (Version Thomas Hartfield) Unit FIVE Page of 39 Ex. 4 Modern smoke detectors use gamma rays produced by Americium-241 to help detect smoke at earlier stages. The half-life of Am-241 is 432 years. If 0.5 grams of Am-241 is used, how much will remain after 10 years when the life of the smoke detector is exhausted (due to mechanical and environmental wear)? Approximate to three decimal places. Ex. 5 Technetium-99m is used in many medical applications including as a tracer for diagnosing bone infections in children. The half-life of Te-99m is 6.02 hours. Riley took a dose of Te-99m at 6am this morning. How much of it remains in her system at 9pm tonight? Express to the tenth of a percent.

38 Hartfield College Algebra (Version Thomas Hartfield) Unit FIVE Page of 39 Ex. 6 A new treatment for cancer that targets bone metastasis involves the use of Radium-223. The half-life of Ra-223 is days. Leslie can only receive new treatments when 90% of the previous application has exited his body. How many days must Leslie wait between treatments? Ex. 7 A cup of coffee brewed at 200ºF is served to a customer at a football game where the temperature is 50ºF. Based on the material of the cup, the cooling rate for the cup is 2 percent per minute. Find the model for the temperature of the coffee, where time is in minutes, and then determine the temperature of the coffee an hour later.

39 Hartfield College Algebra (Version Thomas Hartfield) Unit FIVE Page of 39 Ex. 8 Robin heats water to 190ºF in a kitchen whose temperature is 70ºF. In 6 minutes the temperature of the water is 170ºF. a: Find the formula for the temperature of the water t minutes after it is removed from the stove top. b: In minutes and seconds hsow long will it take the water to cool to 100ºF from the original 190ºF?

Topic 33: One-to-One Functions. Are the following functions one-to-one over their domains?

Topic 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 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

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

4.1 Exponential Functions. For Formula 1, the value of n is based on the frequency of compounding. Common frequencies include:

4.1 Exponential Functions. For Formula 1, the value of n is based on the frequency of compounding. Common frequencies include: Hartfield MATH 2040 Unit 4 Page 1 4.1 Exponential Functions Recall from algebra the formulas for Compound Interest: Formula 1 For Discretely Compounded Interest 1 A t P r n nt Formula 2 Continuously Compounded

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

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

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

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

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

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

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

FLC 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)

FLC 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 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

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

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

Write 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 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 information

Function: State whether the following examples are functions. Then state the domain and range. Use interval notation.

Function: State whether the following examples are functions. Then state the domain and range. Use interval notation. Name Period Date MIDTERM REVIEW Algebra 31 1. What is the definition of a function? Functions 2. How can you determine whether a GRAPH is a function? State whether the following examples are functions.

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

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

8.1 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 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 information

Polynomials 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 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 information

Chapter 7 - Exponents and Exponential Functions

Chapter 7 - Exponents and Exponential Functions Chapter 7 - Exponents and Exponential Functions 7-1: Multiplication Properties of Exponents 7-2: Division Properties of Exponents 7-3: Rational Exponents 7-4: Scientific Notation 7-5: Exponential Functions

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

Math 180 Chapter 4 Lecture Notes. Professor Miguel Ornelas

Math 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 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. Exam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) An initial investment of $14,000 is invested for 9 years in an account

More information

CHAPTER 6. Exponential Functions

CHAPTER 6. Exponential Functions CHAPTER 6 Eponential Functions 6.1 EXPLORING THE CHARACTERISTICS OF EXPONENTIAL FUNCTIONS Chapter 6 EXPONENTIAL FUNCTIONS An eponential function is a function that has an in the eponent. Standard form:

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

7.1 Exponential Functions

7.1 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 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

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

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

MATH 1113 Exam 2 Review

MATH 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 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

Algebra 32 Midterm Review Packet

Algebra 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 information

Review for Final Exam Show your work. Answer in exact form (no rounded decimals) unless otherwise instructed.

Review for Final Exam Show your work. Answer in exact form (no rounded decimals) unless otherwise instructed. Review for Final Eam Show your work. Answer in eact form (no rounded decimals) unless otherwise instructed. 1. Consider the function below. 8 if f ( ) 8 if 6 a. Sketch a graph of f on the grid provided.

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

2(x 4 7x 2 18) 2(x 2 9)(x 2 + 2) 2(x 3)(x + 3)(x 2 + 2)

2(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 information

Lesson 26: Problem Set Sample Solutions

Lesson 26: Problem Set Sample Solutions Problem Set Sample Solutions Problems and 2 provide students with more practice converting arithmetic and geometric sequences between explicit and recursive forms. Fluency with geometric sequences is required

More information

MA Lesson 14 Notes Summer 2016 Exponential Functions

MA 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 information

MATH 1113 Exam 2 Review. Spring 2018

MATH 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 information

OBJECTIVE 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 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 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

2015 2nd Semester Exam Review

2015 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 information

Exponential and Logarithmic Functions

Exponential and Logarithmic Functions Exponential and Logarithmic Functions Lori Jordan Kate Dirga Say Thanks to the Authors Click http://www.ck1.org/saythanks (No sign in required) To access a customizable version of this book, as well as

More information

Simplifying Radical Expressions

Simplifying Radical Expressions Simplifying Radical Expressions Product Property of Radicals For any real numbers a and b, and any integer n, n>1, 1. If n is even, then When a and b are both nonnegative. n ab n a n b 2. If n is odd,

More information

Day Date Assignment. 7.1 Notes Exponential Growth and Decay HW: 7.1 Practice Packet Tuesday Wednesday Thursday Friday

Day 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 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

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

Logarithmic and Exponential Equations and Inequalities College Costs

Logarithmic 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 information

Independent Study Project: Chapter 4 Exponential and Logarithmic Functions

Independent Study Project: Chapter 4 Exponential and Logarithmic Functions Name: Date: Period: Independent Study Project: Chapter 4 Exponential and Logarithmic Functions Part I: Read each section taken from the Algebra & Trigonometry (Blitzer 2014) textbook. Fill in the blanks

More information

#2 Points possible: 1. Total attempts: 2 An exponential function passes through the points (0, 3) and (3, 375). What are the

#2 Points possible: 1. Total attempts: 2 An exponential function passes through the points (0, 3) and (3, 375). What are the Week 9 Problems Name: Neal Nelson Show Scored View #1 Points possible: 1. Total attempts: 2 For each table below, could the table represent a function that is linear, exponential, or neither? f(x) 90 81

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

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

Name Date Per. Ms. Williams/Mrs. Hertel

Name 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 information

Unit 8: Exponential & Logarithmic Functions

Unit 8: Exponential & Logarithmic Functions Date Period Unit 8: Eponential & Logarithmic Functions DAY TOPIC ASSIGNMENT 1 8.1 Eponential Growth Pg 47 48 #1 15 odd; 6, 54, 55 8.1 Eponential Decay Pg 47 48 #16 all; 5 1 odd; 5, 7 4 all; 45 5 all 4

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

4.1 Exponential Functions

4.1 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 information

Summer MA Lesson 20 Section 2.7 (part 2), Section 4.1

Summer 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 information

2. (10 points) Find an equation for the line tangent to the graph of y = e 2x 3 at the point (3/2, 1). Solution: y = 2(e 2x 3 so m = 2e 2 3

2. (10 points) Find an equation for the line tangent to the graph of y = e 2x 3 at the point (3/2, 1). Solution: y = 2(e 2x 3 so m = 2e 2 3 November 24, 2009 Name The total number of points available is 145 work Throughout this test, show your 1 (10 points) Find an equation for the line tangent to the graph of y = ln(x 2 +1) at the point (1,

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

COLLEGE ALGEBRA PRACTICE FINAL (Revised 3/04)

COLLEGE ALGEBRA PRACTICE FINAL (Revised 3/04) Sketch the following graphs:. y x 0 COLLEGE ALGEBRA PRACTICE FINAL (Revised /0) + =. ( ) ( ) f x = x+. ( ) g x = x + 8x 7. y = x. y = x + 6. f ( x) = x + 7. h( x) x + = x + 8. g( x) = x x 9. y = x( x+

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

f(x) = d(x) q(x) + r(x).

f(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 information

Chapter 2 Functions and Graphs

Chapter 2 Functions and Graphs Chapter 2 Functions and Graphs Section 6 Logarithmic Functions Learning Objectives for Section 2.6 Logarithmic Functions The student will be able to use and apply inverse functions. The student will be

More information

DIFFERENTIATION RULES

DIFFERENTIATION RULES 3 DIFFERENTIATION RULES DIFFERENTIATION RULES 3.8 Exponential Growth and Decay In this section, we will: Use differentiation to solve real-life problems involving exponentially growing quantities. EXPONENTIAL

More information

4. Sketch the graph of the function. Ans: A 9. Sketch the graph of the function. Ans B. Version 1 Page 1

4. 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 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

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

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

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

Exponential and Logarithmic Functions

Exponential and Logarithmic Functions Exponential and Logarithmic Functions 6 Figure 1 Electron micrograph of E. Coli bacteria (credit: Mattosaurus, Wikimedia Commons) CHAPTER OUTLINE 6.1 Exponential Functions 6.5 Logarithmic Properties 6.2

More information

Mathematics 5 SN LOGARITHMIC FUNCTIONS. Computers have changed a great deal since 1950 because of the miniaturization of the circuits. on a chip.

Mathematics 5 SN LOGARITHMIC FUNCTIONS. Computers have changed a great deal since 1950 because of the miniaturization of the circuits. on a chip. Mathematics 5 SN LOGARITHMIC FUNCTIONS 1 Computers have changed a great deal since 1950 because of the miniaturization of the circuits. Number of circuits Year on a chip f(x) 1950 1960 1970 1980 1990 35

More information

Exponential Functions

Exponential 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 information

Algebra III: Blizzard Bag #1 Exponential and Logarithm Functions

Algebra 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 information

4.1 Solutions to Exercises

4.1 Solutions to Exercises 4.1 Solutions to Exercises 1. Linear, because the average rate of change between any pair of points is constant. 3. Exponential, because the difference of consecutive inputs is constant and the ratio of

More information

Math-3 Lesson 8-7. b) ph problems c) Sound Intensity Problems d) Money Problems e) Radioactive Decay Problems. a) Cooling problems

Math-3 Lesson 8-7. b) ph problems c) Sound Intensity Problems d) Money Problems e) Radioactive Decay Problems. a) Cooling problems Math- Lesson 8-7 Unit 5 (Part-) Notes 1) Solve Radical Equations ) Solve Eponential and Logarithmic Equations ) Check for Etraneous solutions 4) Find equations for graphs of eponential equations 5) Solve

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

Exponents and Logarithms Exam

Exponents 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 information

EXPONENTS AND LOGS (CHAPTER 10)

EXPONENTS AND LOGS (CHAPTER 10) EXPONENTS AND LOGS (CHAPTER 0) POINT SLOPE FORMULA The point slope formula is: y y m( ) where, y are the coordinates of a point on the line and m is the slope of the line. ) Write the equation of a line

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

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

MATH 1101 Exam 3 Review - Gentry. Spring 2018

MATH 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 information

Algebra II Honors Final Exam Review

Algebra II Honors Final Exam Review Class: Date: Algebra II Honors Final Exam Review Short Answer. Evaluate the series 5n. 8 n =. Evaluate the series (n + ). n = What is the sum of the finite arithmetic series?. 9+ + 5+ 8+ + + 59. 6 + 9

More information

MAT 111 Final Exam Fall 2013 Name: If solving graphically, sketch a graph and label the solution.

MAT 111 Final Exam Fall 2013 Name: If solving graphically, sketch a graph and label the solution. MAT 111 Final Exam Fall 2013 Name: Show all work on test to receive credit. Draw a box around your answer. If solving algebraically, show all steps. If solving graphically, sketch a graph and label the

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

Two-Year Algebra 2 A Semester Exam Review

Two-Year Algebra 2 A Semester Exam Review Semester Eam Review Two-Year Algebra A Semester Eam Review 05 06 MCPS Page Semester Eam Review Eam Formulas General Eponential Equation: y ab Eponential Growth: A t A r 0 t Eponential Decay: A t A r Continuous

More information

CHAPTER 7. Logarithmic Functions

CHAPTER 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

Exponential and Logarithmic Equations

Exponential and Logarithmic Equations OpenStax-CNX module: m49366 1 Exponential and Logarithmic Equations OpenStax College This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 4.0 In this section,

More information

MATH 112 Final Exam Study Questions

MATH 112 Final Exam Study Questions MATH Final Eam Study Questions Spring 08 Note: Certain eam questions have been more challenging for students. Questions marked (***) are similar to those challenging eam questions.. A company produces

More information

Algebra 2 - Semester 2 - Final Exam Review

Algebra 2 - Semester 2 - Final Exam Review Algebra 2 - Semester 2 - Final Exam Review Your final exam will be 60 multiple choice questions coving the following content. This review is intended to show examples of problems you may see on the final.

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

Section 6.8 Exponential Models; Newton's Law of Cooling; Logistic Models

Section 6.8 Exponential Models; Newton's Law of Cooling; Logistic Models Section 6.8 Exponential Models; Newton's Law of Cooling; Logistic Models 197 Objective #1: Find Equations of Populations that Obey the Law of Uninhibited Growth. In the last section, we saw that when interest

More information

COLLEGE ALGEBRA. Practice Problems Exponential and Logarithm Functions. Paul Dawkins

COLLEGE ALGEBRA. Practice Problems Exponential and Logarithm Functions. Paul Dawkins COLLEGE ALGEBRA Practice Problems Eponential and Logarithm Functions Paul Dawkins Table of Contents Preface... ii Eponential and Logarithm Functions... Introduction... Eponential Functions... Logarithm

More information

ALGEBRA 2/TRIGONMETRY TOPIC REVIEW QUARTER 2 POWERS OF I

ALGEBRA 2/TRIGONMETRY TOPIC REVIEW QUARTER 2 POWERS OF I ALGEBRA /TRIGONMETRY TOPIC REVIEW QUARTER Imaginary Unit: i = i i i i 0 = = i = = i Imaginary numbers appear when you have a negative number under a radical. POWERS OF I Higher powers if i: If you have

More information

Solving Exponential Equations (Applied Problems) Class Work

Solving Exponential Equations (Applied Problems) Class Work Solving Exponential Equations (Applied Problems) Class Work Objective: You will be able to solve problems involving exponential situations. Quick Review: Solve each equation for the variable. A. 2 = 4e

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

Geometry Placement Exam Review Revised 2017 Maine East High School

Geometry 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 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

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