Recognize the appropriate method for factoring a polynomial Use a general strategy for factoring polynomials

Transcription

1 Section 6.5: A GENERAL FACTORING STRATEGY When you are done with your homework you should be able to Recognize the appropriate method for factoring a polynomial Use a general strategy for factoring polynomials WARM-UP: Multiply: a. x 1 x x 1 b. x 3y4x 6xy 9y A STRATEGY FOR FACTORING A POLYNOMIAL 1. If there is a factor other than, factor the.. Determine the of in the polynomial and try factoring as follows: a. If there are terms, can the be factored by one of the following special forms? of : 1

2 of : of : b. If there are terms, is the a? If so, factor by one of the following special forms: = = If the trinomial is a, try by and or. c. If there are or terms, try by. 3. Check to see if any with more than one term in the can be factored. If so, completely. 4. by.

3 Example 1: Factor a. 5x 45x 4 b. 4x 16x 48 c. 5 4x 64 x d. 3 x x x

4 e x x x f. w 54w 5 g. 3x y 48y 4 5 h. 1x 36x y 7xy 3 4

5 i. 1x x 1 4x x 1 5 x 1 j. x 14x 49 16a APPLICATION Express the area of the shaded ring shown in the figure in terms of π. Then factor this expression completely. b a 5

6 Section 6.6: SOLVING QUADRATIC EQUATIONS BY FACTORING When you are done with your homework you should be able to Use the zero-product principle Solve quadratic equations by factoring Solve problems using quadratic equations WARM-UP: a. Factor: x 8x 7 b. Solve: x 7 0 DEFINITION OF A QUADRATIC EQUATION A in is an equation that can be written in the where,, and are real numbers, with. A in is also called a - equation in. 6

7 SOLVING QUADRATIC EQUATIONS BY FACTORING Consider the quadratic equation first warm-up? x 8x 7 0. How is this different from the We can the side of the equation to get. If a quadratic equation has a zero on one side and a on the other side, it can be using the - principle. THE ZERO-PRODUCT PRINCIPLE If the of two or more expressions is, then one of them is to. 7

8 Example 1: Solve the following equations: a. x 11 0 b. x 1 0 c. x x STEPS FOR SOLVING A QUADRATIC EQUATION BY FACTORING 1. If necessary, the equation in form, moving all to one side, thereby obtaining on the other side Apply the - principle, setting each equal to. 4. the equations formed in step the in the equation. 8

9 Example : Solve: a. x x 9 0 b. x x c. x x 4 0 d. x 8x 9

10 e. 4x 1x 9 f. x x g. 3 x 4x 0 10

11 h. x x APPLICATION An explosion causes debris to rise vertically with an initial velocity of 7 feet per second. The formula h 16t 7t describes the height of the debris above the ground, h, in feet, t seconds after the explosion. a. How long will it take for the debris to hit the ground? b. When will the debris be 3 feet above the ground? 11

12 Section 7.1: RATIONAL EXPRESSIONS AND THEIR SIMPLIFICATION When you are done with your homework you should be able to Find numbers for which a rational expression is undefined Simplify rational expressions Solve applied problems involving rational expressions WARM-UP: a. Factor: 3 x x x 8 16 b. Solve: x x 10 0 EXCLUDING NUMBERS FROM RATIONAL EXPRESSIONS A expression is the of two. Rational expressions indicate and division by is. This means that we any value or values of the that make a! 1

13 Example 1: Find all numbers for which the rational expression is undefined: a. 5 x b. x 1 x 4 8x 40 c. x 3x 8 x 1 d. x 4 SIMPLIFYING RATIONAL EXPRESSIONS A is if its and have common other than or. FUNDAMENTAL PRINCIPLE OF RATIONAL EXPRESSIONS If,, and are and and are, 13

14 STEPS FOR SIMPLIFYING RATIONAL EXPRESSIONS 1. the and the completely.. both the and the by any. Example : Simplify: a. 4 x 64 16x b. 6y 18 11y 33 c. x 1x 36 4x 4 14

15 d. 3 x x x x 4 e. x 5 x 5 f. 3 x 1 x 1 15

16 SIMPLIFYING RATIONAL EXPRESIONS WITH OPPOSITE FACTORS IN THE NUMERATOR AND DENOMINATOR The of two that have signs and are is. Example 3: Simplify: a. x 3 3 x b. 9 x x c. x 4 x 16

17 APPLICATION A company that manufactures small canoes has costs given by the equation C 0x 0000 x in which x is the number of canoes manufactured and C is the cost to manufacture each canoe. a. Find the cost per canoe when manufacturing 100 canoes. b. Find the cost per canoe when manufacturing canoes. c. Does the cost per canoe increase or decrease as more canoes are manufactured? 17

18 Section 7.: MULTIPLYING AND DIVIDING RATIONAL EXPRESSIONS When you are done with your homework you should be able to Multiply rational expressions Divide rational expressions WARM-UP: Simplify: a. a ab b a b b. x 3x x 1 MULTIPLYING RATIONAL EXPRESSIONS If,,, and are polynomials, where and, then The of two is the of their, divided by the of their. 18

19 STEPS FOR MULTIPLYING RATIONAL EXPRESSIONS 1. all and.. and by common. 3. the remaining factors in the and the remaining factors in the. Example 1: Multiply. a. x x 8 9y 1 y c. y y 3y 7 b. x 30 5 x 4 d. x 5x 6 x 9 x x 6 x x 6 19

20 DIVIDING RATIONAL EXPRESSIONS If,,, and are polynomials, where,, and, then The of two is the of the expression and the of the. Example : Divide. a. x c. y y y 15 5 b. x 5 4x d. x 4y x 4xy 4y x 3xy y x y 0

21 Example 3: Perform the indicated operation or operations. a. 5x x 6x x x x 1 3x 10x 3x 1 x x b xy ay 5xb ab y b 5x a 15x 3a 1

22 Section 7.3: ADDING AND SUBTRACTING RATIONAL EXPRESSIONS WITH THE SAME DENOMINATOR When you are done with your homework you should be able to Add rational expressions with the same denominator Subtract rational expressions with the same denominator Add and subtract rational expressions with opposite denominators WARM-UP: Simplify: a. b a a b b. x x 1 1 x ADDING RATIONAL EXPRESSIONS WITH COMMON DENOMINATORS If and are expressions, then To rational expressions with the, add and place the over the. If possible, the result.

23 SUBTRACTING RATIONAL EXPRESSIONS WITH COMMON DENOMINATORS If and are expressions, then To rational expressions with the, subtract and place the over the. If possible, the result. Example 1: Add or subtract as indicated. Simplify the result, if possible. a. x 4x c. x 1 x 1 x 1 b. x 4 x x 3 x 6 d. 3x 4 3x 4 3

24 e. 3 3 x 3 7x x x f. x 9x 3x 5x 4x 11x 3 4x 11x 3 g. 3y y 10 y 6y 3y 10y 8 3y 10y 8 3y 10y 8 4

25 ADDING AND SUBTRACTING RATIONAL EXPRESSIONS WITH OPPOSITE DENOMINATORS When one denominator is the, or, of the other, first either rational expression by to obtain a. Example : Add or subtract as indicated. Simplify the result, if possible. 6 x 7 3 x a. x 6 6 x 4 x 3 x 8 c. x 9 9 x b. x 9 x 3 3 x x 3 x x x x x d. 5

26 Section 7.4: ADDING AND SUBTRACTING RATIONAL EXPRESSIONS WITH DIFFERENT DENOMINATORS When you are done with your homework you should be able to Find the least common denominator Add and subtract rational expressions with different denominators WARM-UP: Perform the indicated operation and simplify. a. 3 5 x 1 b. 8 1 x x x x FINDING THE LEAST COMMON DENOMINATOR (LCD) The denominator of several is a consisting Of the of all in the, with each raised to the greatest of its occurrence in any denominator. CREATED BY SHANNON MARTIN GRACEY 6

27 FINDING THE LEAST COMMON DENOMINATOR 1. each completely.. List the factors of the first. 3. Add to the list in step any of the second denominator that do not appear in the list. Repeat this step for all denominators. 4. Form the of the from the list in step 3. This product is the LCD. Example 1: Find the LCD of the rational expressions a. and 5x 35x 7 1 b. and y 49 y 14y 49 ADDING AND SUBTRACTING RATIONAL EXPRESSIONS THAT HAVE DIFFERENT DENOMINATORS 1. Find the of the.. Rewrite each rational expression as an expression whose is the. 3. Add or subtract, placing the resulting expression over the LCD. 4. If possible, the resulting rational expression. CREATED BY SHANNON MARTIN GRACEY 7

28 Example : Add or subtract as indicated. Simplify the result, if possible. a x 8x b. 1 3 x c. x 3x x 3 d. y y 5 y 5 y CREATED BY SHANNON MARTIN GRACEY 8

29 3x 7 3 e. x 5x 6 x 3 f. 5 3 x 36 6 x ADDING AND SUBTRACTING RATIONAL EXPRESSIONS WHEN DENOMINATORS CONTAIN OPPOSITE FACTORS When one denominator contains the factor of the other, first either rational expression by. Then apply the for or rational expressions that have. CREATED BY SHANNON MARTIN GRACEY 9

30 Example 3: Add or subtract as indicated. Simplify the result, if possible. x 7 x a. 4x 1 9 x 5 x x y y x b. 7 y y y 1 c. y y 1 4 y y 3 CREATED BY SHANNON MARTIN GRACEY 30

31 Section 7.5: COMPLEX RATIONAL EXPRESSIONS When you are done with your homework you should be able to Simplify complex rational expressions by dividing Simplify complex rational expressions by multiplying by the LCD WARM-UP: Perform the indicated operation. Simplify, if possible. a. x 1 3x x x 1 b. x x 1x x 4 x 4 SIMPLIFYING A COMPLEX RATIONAL EXPRESSION BY DIVIDING 1. If necessary, add or subtract to get a rational expression in the.. If necessary, add or subtract to get a rational expression in the. 3. Perform the indicated by the main bar: the denominator of the complex rational expression and. 4. If possible,. CREATED BY SHANNON MARTIN GRACEY 31

32 Let s simplify the problem below using this method: Now let s replace the constants with variables and simplify using the same method. 1 x x 1 4 x 1 CREATED BY SHANNON MARTIN GRACEY 3

33 Example 1: Simplify each complex rational expression. a x x b. 1 1 x y 1 1 x y CREATED BY SHANNON MARTIN GRACEY 33

34 c. 8 x x 10 6 x x d. 1 x 1 1 x CREATED BY SHANNON MARTIN GRACEY 34

35 SIMPLIFYING A COMPLEX RATIONAL EXPRESSION BY MULTIPLYING BY THE LCD 1. Find the LCD of ALL expressions within the rational expression.. both the and by this LCD. 3. Use the property and multiply each in the numerator and denominator by this. each term. No expressions should remain. 4. If possible, and. Let s simplify the earlier problem using this method: CREATED BY SHANNON MARTIN GRACEY 35

36 Now let s replace the constants with variables and simplify using the same method. 1 x x 1 4 x 1 Example : Simplify each complex rational expression. a. 7 4 y 3 y CREATED BY SHANNON MARTIN GRACEY 36

37 b. 3 x x 3 x 3 3 x c. 5 x y xy x y xy 3 4 CREATED BY SHANNON MARTIN GRACEY 37

38 d. 1 x 1 1 x Example 3: Simplify each complex rational expression using the method of your choice. a. 3 3 x x 5 x 4 CREATED BY SHANNON MARTIN GRACEY 38

39 b. y y 1 1 Application: The average rate on a round-trip commute having a one-way distance d is given by the complex rational expression d d d r r 1 on the outgoing and return trips, respectively. a. Simplify the expression. in which r 1 and r are the average rates b. Find your average rate if you drive to the campus averaging 40 mph and return home on the same route averaging 30 mph. CREATED BY SHANNON MARTIN GRACEY 39

40 Section 7.6: SOLVING RATIONAL EQUATIONS When you are done with your homework you should be able to Solve rational equations Solve problems involving formulas with rational expressions Solve a formula with a rational expression for a variable WARM-UP: Solve. 3x x 8 0 SOLVING RATIONAL EQUATIONS 1. List on the variable. (Remember no in the denominator!). Clear the equation of fractions by multiplying sides of the equation by the LCD of rational expressions in the equation. 3. the resulting equation. 4. Reject any proposed solution that is in the list of on the variable. other proposed solutions in the equation. CREATED BY SHANNON MARTIN GRACEY 40

41 Example 1: Solve each rational equation. a. 7 5 x 3x 3 b. 10 5y 3 y y CREATED BY SHANNON MARTIN GRACEY 41

42 c. x 1 6 x 3 x t t 1 6t 8 t t 1 t t t t t d. 3 CREATED BY SHANNON MARTIN GRACEY 4

43 e. 3y 1 4y 1 SOLVING A FORMULA FOR A VARIABLE Formulas and models frequently contain rational expressions. The goal is to get the variable on one side of the equation. It is sometimes necessary to out the variable you are solving for. Example : Solve each formula for the specified variable. a. V V P P 1 1 for V x x b. z for x s c. f f1 f f f 1 for f CREATED BY SHANNON MARTIN GRACEY 43

44 Section 7.7: APPLICATIONS USING RATIONAL EQUATIONS AND PROPORTIONS When you are done with your homework you should be able to Solve problems involving motion Solve problems involving work Solve problems involving proportions Solve problems involving similar triangles WARM-UP: A motorboat traveled 36 miles downstream, with the current, in 1.5 hours. The return trip upstream, against the current, covered the same distance, but took hours. Find the boat s rate in still water and the rate of the current. CREATED BY SHANNON MARTIN GRACEY 44

45 PROBLEMS INVOLVING MOTION Recall that. Rational expressions appear in problems when the conditions of the problem involve the traveled. When we isolate time in the formula above, we get Example 1: As part of an exercise regimen, you walk miles on an indoor track. Then you jog at twice your walking speed for another miles. If the total time spent walking and jogging is 1 hour, find the walking and jogging rates. CREATED BY SHANNON MARTIN GRACEY 45

46 Example : The water s current is mph. A canoe can travel 6 miles downstream, with the current, in the same amount of time that it travels miles upstream, against the current. What is the canoe s average rate in still water? PROBLEMS INVOLVING WORK In problems, the number represents one job. Equations in work problems are based on the following condition: CREATED BY SHANNON MARTIN GRACEY 46

47 Example 3: Shannon can clean the house in 4 hours. When she worked with Rory, it took 3 hours. How long would it take Rory to clean the house if he worked alone? Example 4: A hurricane strikes and a rural area is without food or water. Three crews arrive. One can dispense needed supplies in 10 hours, a second in 15 hours, and a third in 0 hours. How long will it take all three crews working together to dispense food and water? CREATED BY SHANNON MARTIN GRACEY 47

48 PROBLEMS INVOLVING PROPORTIONS A ratio is the quotient of two numbers or two quantities. The ratio of two numbers a and b can be written as a to b a : b or a b or A proportion is an equation of the form, a b c where b 0 and d 0. We call a, d b, c, and d the terms of the proportion. The cross-products ad and bc are equal. Example 5: According to the authors of Number Freaking, in a global village of 00 people, 9 get drunk every day. How many of the world s 6.9 billion people (010 population) get drunk every day? Example 6: A person s hair length is proportional to the number of years it has been growing. After years, a person s hair grows 8 inches. The longest moustache on record was grown by Kalyan Sain of India. Sain grew his moustache for 17 years. How long was each side of the moustache? CREATED BY SHANNON MARTIN GRACEY 48

49 SIMILAR FIGURES Two figures are similar if their corresponding angle measures are equal and their corresponding sides are proportional. Example 7: A fifth-grade student is conducting an experiment to find the height of a tree in the schoolyard. The student measures the length of the tree s shadow and then immediately measures the length of the shadow that a yardstick forms. The tree s shadow measures 30 feet and the yardstick s shadow measures 6 feet. Find the height of the tree. CREATED BY SHANNON MARTIN GRACEY 49

50 Section 8.1: INTRODUCTION TO FUNCTIONS When you are done with your homework you should be able to Find the domain and range of a relation Determine whether a relation is a function Evaluate a function WARM-UP: Evaluate y x x 5 at x 3. DEFINITION OF A RELATION A is any of ordered pairs. The set of all components of the pairs is called the of the relation and the set of all second components is called the of the. Example 1: Find the domain and range of the relation. VEHICLE NUMBER OF WHEELS CAR 4 MOTORCYCLE BOAT 0 CREATED BY SHANNON MARTIN GRACEY 50

51 DEFINITION OF A FUNCTION A is a from a first set, called the, to a second set, called the, such that each in the corresponds to element in the. Example : Determine whether each relation represents a function. Then identify the domain and range. a. 6,1, 1,1, 0,1, 1,1,,1 b. 3,3,,0, 4,0,, 5 CREATED BY SHANNON MARTIN GRACEY 51

52 FUNCTIONS AS EQUATIONS AND FUNCTION NOTATION Functions are often given in terms of rather than as of. Consider the equation below, which describes the position of an object, in feet, dropped from a height of 500 feet after x seconds. y 16x 500 The variable is a of the variable. For each value of x, there is one and only one value of. The variable x is called the variable because it can be any value from the. The variable y is called the variable because its value on x. When an represents a, the function is often named by a letter such as f, g, h, F, G, or H. Any letter can be used to name a function. The domain is the of the function s and the range is the of the function s. If we name our function, the input is represented by, and the output is represented by. The notation is read of or at. So we may rewrite y 16x 500 as. Now let s evaluate our function after 10 seconds: CREATED BY SHANNON MARTIN GRACEY 5

53 f x x x x 10. Example 3: Find the indicated function values for 3 a. f 0 b. f c. f d. f 1 f 1 Example 3: Find the indicated function and domain values using the table below. a. h x h x b. h 1 c. For what values of x is h x 1? CREATED BY SHANNON MARTIN GRACEY 53

54 Section 8.: GRAPHS OF FUNCTIONS When you are done with your homework you should be able to Use the vertical line test to identify functions Obtain information about a function from its graph Review interval notation Identify the domain and range of a function from its graph WARM-UP: Graph the following equations by plotting points. a. y x b. y 3x 1 CREATED BY SHANNON MARTIN GRACEY 54

55 THE VERTICAL LINE TEST FOR FUNCTIONS If any vertical line a graph in more than point, the graph define as a function of. Example 1: Determine whether the graph is that of a function. a. b. c. OBTAINING INFORMATION FROM GRAPHS You can obtain information about a function from its graph. At the right or left of a graph, you will often find dots, dots, or. A closed dot indicates that the graph does not beyond this point and the belongs to the An open dot indicates that the graph does not beyond this point and the DOES NOT belong to the CREATED BY SHANNON MARTIN GRACEY 55

56 An arrow indicates that the graph extends in the direction in which the arrow REVIEWING INTERVAL NOTATION INTERVAL NOTATION a, b a, b a, b a, b a, a,,b,b, SET-BUILDER NOTATION GRAPH x x x x x x x x x CREATED BY SHANNON MARTIN GRACEY 56

57 Example : Use the graph of f to determine each of the following. f a. f 0 b. f c. For what value of x is f x 3? d. The domain of f e. The range of f CREATED BY SHANNON MARTIN GRACEY 57

58 Example 3: Graph the following functions by plotting points and identify the domain and range. a. f x x b. H x x 1 CREATED BY SHANNON MARTIN GRACEY 58

59 Section 8.3: THE ALGEBRA OF FUNCTIONS When you are done with your homework you should be able to Find the domain of a function Use the algebra of functions to combine functions and determine domains WARM-UP: Find the following function values for a. f 4 f x x b. f 0 c. f 196 FINDING A FUNCTION S DOMAIN If a function f does not model data or verbal conditions, its domain is the set of numbers for which the value of f x is a real number. from a function s real numbers that cause by and real numbers that result in a root of a number. Example 1: Find the domain of each of the following functions. a. f x x 1 4 x 1 x b. g x c. ht 3t 5 CREATED BY SHANNON MARTIN GRACEY 59

60 THE ALGEBRA OF FUNCTIONS Consider the following two functions: f x x and g x 3x 5 Let s graph these two functions on the same coordinate plane. Now find and graph the sum of f and g. f g x CREATED BY SHANNON MARTIN GRACEY 60

61 Now find and graph the difference of f and g. f g x Now find and graph the product of f and g. fg x Now find and graph the quotient of f and g. f g x CREATED BY SHANNON MARTIN GRACEY 61

62 THE ALGEBRA OF FUNCTIONS: SUM, DIFFERENCE, PRODUCT, AND QUOTIENT OF FUNCTIONS Let f and g be two functions. The f the fg, and the g, the f g f are whose g domains are the set of all real numbers to the domains of f and g, defined as follows: 1. Sum:,. Difference: 3. Product: 4. Quotient:, provided Example : Let f x x 4x and g x x. Find the following: a. f g x d. fg x b. f g 4 e. fg 3 c. f 3 g 3 f g f. The domain of x CREATED BY SHANNON MARTIN GRACEY 6

63 Section 8.4: COMPOSITE AND INVERSE FUNCTIONS When you are done with your homework you should be able to Form composite functions Verify inverse functions Find the inverse of a function Use the horizontal line test to determine if a function has an inverse function Use the graph of a one-to-one function to graph its inverse function WARM-UP: Find the domain and range of the function 1,0, 0,1, 1,,,3 : THE COMPOSITION OF FUNCTIONS The composition of the function with is denoted by and is defined by the equation The domain of the function is the set of all such that 1. is in the domain of and. is in the domain of. CREATED BY SHANNON MARTIN GRACEY 63

64 Example 1: Given f x x 8 and g x 6x 1 composite functions. a. f g x b. g f x, find each of the following DEFINITION OF THE INVERSE OF A FUNCTION Let f and g be two functions such that for every in the domain of and for every in the domain of. The function is the of the function and is denoted by (read f -inverse ). Thus and. The of is equal to the of and vice versa. CREATED BY SHANNON MARTIN GRACEY 64

65 Example : Show that each function is the inverse of the other. f x 4x 9 and g x x 9 4 FINDING THE INVERSE OF A FUNCTION The equation of the inverse of a function f can be found as follows: 1. Replace with in the equation for.. Interchange and. 3. Solve for. If this equation does not define as a function of, the function doe not have an function and this procedure ends. If this equation does define as a function of, the function has an inverse function. 4. If has an inverse function, replace in step 3 with. We can verify our result by showing that and. CREATED BY SHANNON MARTIN GRACEY 65

66 Example 3: Find an equation for f 1 x, the inverse function. a. f x 4x b. f x x 3 x 1 THE HORIZONTAL LINE TEST FOR INVERSE FUNCTIONS A function f has an inverse that is a function, if there is no line that intersects the graph of the function at more than point. Example 4: Which of the following graphs represent functions that have inverse functions? a. b. c. CREATED BY SHANNON MARTIN GRACEY 66

67 GRAPHS OF A FUNCTION AND ITS INVERSE FUNCTION There is a between the graph of a one-to-one function and its inverse. Because inverse functions have ordered pairs with the coordinates, if the point is on the graph of, the point is on the graph of. The points and are with respect to the line. Therefore, the graph of is a of the graph of about the line. Example 5: Use the graph of f below to draw the graph of its inverse function. 67

68 Section 9.3: EQUATIONS AND INEQUALITIES INVOLVING ABSOLUTE VALUE When you are done with your homework you should be able to Solve absolute value equations Solve absolute value inequalities in the form u Solve absolute value inequalities in the form u Recognize absolute value inequalities with no solution or all real numbers as solutions Solve problems using absolute value inequalities WARM-UP: Graph the solutions of the inequality. a. 6 x 6 c c REWRITING AN ABSOLUTE VALUE EQUATION WITHOUT ABSOLUTE VALUE BARS If is a positive real number and represents any expression, then is equivalent to or. Consider x 6. 68

69 Now consider x 3 6. Example 1: Solve. a. 5x 7 1 b. 7 x 11 1 c. x d. x

70 REWRITING AN ABSOLUTE VALUE EQUATION WITH TWO ABSOLUTE VALUES WITHOUT ABSOLUTE VALUE BARS If, then or. Example : Solve. x 7 x 1 SOLVING ABSOLUTE VALUE INEQUALITIES OF THE FORM u c If is a positive real number and represents any expression, then This rule is valid if is replaced by. Example 3: Solve and graph the solution set on a number line: a. x 6 70

71 b. 3 x SOLVING ABSOLUTE VALUE INEQUALITIES OF THE FORM u c If is a positive real number and represents any expression, then This rule is valid if is replaced by. Example 4: Solve and graph the solution set on a number line: a. x 6 71

72 b. 5 1x 1 10 ABSOLUTE VALUE INEQUALITIES WITH UNUSUAL SOLUTION SETS If algebraic expression and is a number, 1. The inequality has solution.. The inequality is for all real numbers for which is defined. APPLICATION The inequality 50 T describes the range of monthly average temperature T, in degrees Fahrenheit, for Albany, New York. Solve the inequality and interpret the solution. 7

73 Section 10.1: RADICAL EXPRESSIONS AND FUNCTIONS When you are done with your homework you should be able to Evaluate square roots Evaluate square root functions Find the domain of square root functions Use models that are square root functions Simplify expressions of the form Evaluate cube root functions Simplify expressions of the form 3 a 3 Find even and odd roots Simplify expressions of the form n WARM-UP: a a n 1. Fill in the blank. a. 5 5 d. 16 b. x x 3 6 e. 16 c. y y 16. Solve x Graph f x x 73

74 DEFINITION OF THE PRINCIPAL SQUARE ROOT If is a nonnegative real number, the number such that, denoted by, is the of. Example 1: Evaluate. a. 169 b c d e SQUARE ROOT FUNCTIONS Because each number,, has precisely one principal square root,, there is a square root function defined by The domain of this function is. We can graph by selecting nonnegative real numbers for. It is easiest to pick perfect. How is this different than the graph we sketched in the warm-up? 74

75 Example : Find the indicated function value. a. f x 6x 10; f 1 b. g x 50 x; f 5 Example 3: Find the domain of f x 10x 7 SIMPLIFYING a For any real number a, In words, the principal square root of is the of. Example 4: Simplify each expression. a. 9 c x b. x 3 d. x 14x 49 75

76 DEFINITION OF THE CUBE ROOT OF A NUMBER The cube root of a real number a is written. means that. CUBE ROOT FUNCTIONS Unlike square roots, the cube root of a negative number is a number. All real numbers have cube roots. Because every number,, has precisely one cube root,, there is a cube root function defined by The domain of this function is. We can graph by selecting real numbers for. It is easiest to pick perfect. SIMPLIFYING 3 a 3 For any real number a, In words, the cube root of any expression is that expression. 76

77 Example 5: Find the indicated function value. a. f x 3 x 0; f 1 b. g x 3 x; g 3 Example 6: Graph the following functions by plotting points. a. f x x 1 x f x x 1 x, y g x b. 3 x 3 x g x x x, y 77

78 SIMPLIFYING n a n For any real number a, 1. If n is even,.. If n is odd,. Example 7: Simplify. a. 6 6 x 5 b. x c. 8 APPLICATION Police use the function f x 0x to estimate the speed of a car, f x, in miles per hour, based on the length, x, in feet, of its skid marks upon sudden braking on a dry asphalt road. A motorist is involved in an accident. A police officer measures the car s skid marks to be 45 feet long. If the posted speed limit is 35 miles per hour and the motorist tells the officer she was not speeding, should the officer believe her? 78

79 Section 10.: RATIONAL EXPONENTS When you are done with your homework you should be able to Use the definition of Use the definition of 1 n a m n a m a n Use the definition of Simplify expressions with rational exponents Simplify radical expressions using rational exponents WARM-UP: x y 5. Simplify 3 3 x THE DEFINITION OF 1 n a If represents a real number and is an integer, then If n is even, a must be. If n is odd, a can be any real number. 79

80 Example 1: Use radical notation to rewrite each expression. Simplify, if possible. a b xy c. 1 5 Example : Rewrite with rational exponents. a. 4 1st b z c. 5xyz THE DEFINITION OF m n a If represents a real number, is a positive rational number reduced to lowest terms, and is an integer, then and 80

81 Example 3: Use radical notation to rewrite each expression. Simplify, if possible. a b c. 5 9 Example 4: Rewrite with rational exponents. a b. 5 x y 4 c. 11t 3 THE DEFINITION OF m a n If is a nonzero real number, then 81

82 Example 5: Rewrite each expression with a positive exponent. Simplify, if possible. a b. 3 c PROPERTIES OF RATIONAL EXPONENTS If m and n are rational exponents, and a and b are real numbers for which the following expressions are defined, then 1. m n b b.. b b m n. m 3. b n 4. ab n.. 5. a b n. 8

83 Example 6: Use properties of rational exponents to simplify each expression. Assume that all variables represent positive numbers. a b x y 3 c. y y SIMPLIFYING RADICAL EXPRESSIONS USING RATIONAL EXPONENTS 1. Rewrite each radical expression as an expression with a.. Simplify using of rational exponents. 3. in radical notation if rational exponents still appear. 83

84 Example 7: Use rational exponents to simplify. If rational exponents appear after simplifying, write the answer in radical notation. Assume that all variables represent positive numbers. a. 1 3 xy b c a b ab 84

85 Section 10.3: MULTIPLYING AND SIMPLIFYING RADICAL EXPRESSIONS When you are done with your homework you should be able to Use the product rule to multiply radicals Use factoring and the product rule to simplify radicals Multiply radicals and then simplify WARM-UP: 1. Use properties of rational exponents to simplify each expression. Assume that all variables represent positive numbers. a b x y. Factor out the greatest common factor x 16x 3. Multiply 1 3 x 3 x 10 85

86 THE PRODUCT RULE FOR RADICALS If n a and n b are real numbers, then The of two is the root of the of the radicals. Example 1: Multiply. a. 11 b. 3 4x 3 1x c. x 1 x 1 SIMPLIFYING RADICAL EXPRESSIONS BY FACTORING A radical expression whose index is n is when its radicand has no that are perfect powers. To simplify, use the following procedure: 1. Write the radicand as the of two factors, one of which is the perfect power.. Use the rule to take the root of each factor. 3. Find the root of the perfect nth power. 86

87 Example : Simplify by factoring. Assume that all variables represent positive numbers. a. 1 b. 3 81x 5 c x y z **For the remainder of this chapter, in situations that do not involve functions, we will assume that no radicands involve negative quantities raised to even powers. Based upon this assumption, absolute vale bars are not necessary when taking even roots. SIMPLIFYING WHEN VARIABLES TO EVEN POWERS IN A RADICAND ARE NONNEGATIVE QUANTITIES For any real number a, Example 3: Simplify. a x y b x y z c x y 87

88 Example 4: Multiply and simplify. a. 15xy 3xy b. 10x y 00x y 3 3 Example 5: Simplify. a. 5xy 10xy b. 8x y z 8xy z c. x y 4 8xy 3xy 4 x y 3 88

89 Section 10.4: ADDING, SUBTRACTING, AND DIVIDING RADICAL EXPRESSIONS When you are done with your 10.4 homework you should be able to Add and subtract radical expressions Use the quotient rule to simplify radical expressions Use the quotient rule to divide radical expressions WARM-UP: Simplify. a x y x y b. 3xy 16x y 3 THE QUOTIENT RULE FOR RADICALS If n a and n b are real numbers, and, then The root of a is the of the roots of the and. Example 1: Simplify using the quotient rule. a. 0 9 b. 3 6 x 7 y 1 89

90 DIVIDING RADICAL EXPRESSIONS If n a and n b are real numbers, and, then To two radical expressions with the SAME, divide the radicands and retain the. Example : Divide and, if possible, simplify. a. 10x 3x 4 b. 3 18x y 3 xy 4 4 Example 3: Perform the indicated operations. a. 5 c b. 3 0x 3x 80x d x 3x y 4 x x x y x y

91 10.5: MULTIPLYING WITH MORE THAN ONE TERM AND RATIONALIZING DENOMINATORS When you are done with your 10.5 homework you should be able to Multiply radical expressions with more than one term Use polynomial special products to multiply radicals Rationalize denominators containing one term Rationalize denominators containing two terms Rationalize numerators WARM-UP: Multiply. 1 a. x x 3 b. x 5x 5 c. 3x 1 MULTIPLYING RADICAL EXPRESSIONS WITH MORE THAN ONE TERM Radical expressions with more than one term are multiplied in much the same way as with more than one term are multiplied. Example 1: Multiply. a. 5 x 10 c b. 3 y 16 3 y 91

92 Example : Multiply. a. x 10 x 10 b. a b a b c CONJUGATES Radical expressions that involve the and of the two terms are called. 9

93 RATIONALIZING DENOMINATORS CONTAINING ONE TERM When you a radical expression as an expression in which the denominator no longer contains any. When the denominator contains a radical with an nth root, multiply the and the by a radical of index n that produces a perfect power in the denominator s radicand. Example 3: Rationalize each denominator. a. 3 c. 5 6xy b x d xy 93

94 RATIONALIZING DENOMINATORS CONTAINING TWO TERMS When the denominator contains two terms with one or more roots, multiply the and the by a by the of the denominator. Example 4: Rationalize each denominator. a b c d. x 8 x 3 94

95 RATIONALIZING NUMERATORS To rationalize a numerator, multiply by to eliminate the radical in the. Example 5: Rationalize each numerator. a. 3 b. 5x 4 3 c. x x 95

96 Section 10.6: RADICAL EQUATIONS When you are done with your homework you should be able to Solve radical equations Use models that are radical functions to solve problems WARM-UP: Solve: x 3x 5 SOLVING RADICAL EQUATIONS CONTAINING nth ROOTS 1. If necessary, arrange terms so that radical is on one side of the equation.. Raise sides of the equation to the power to eliminate the nth root. 3. the resulting equation. If this equation still contains radicals, steps 1 and! 4. all proposed solutions in the equation. 96

97 Example 1: Solve. a. 5x 1 8 b. x c. x 6x 7 d. 3 4x

98 e. x 3x 7 1 f. x 3 4 x 1 1 g. 1 x 1 3 x x 3 98

99 Example : If f x x x, find all values of x for which f x 4. Example 3: Solve A r for A. 4 99

100 Example 4: Without graphing, find the x -intercept of the function f x x 3 x 1. APPLICATION A basketball player s hang time is the time spent in the air when shooting a d basket. The formula t models hang time, t, in seconds, in terms of the vertical distance of a player s jump, d, in feet. When Michael Wilson of the Harlem Globetrotters slam-dunked a basketball 1 feet, his hang time for the shot was approximately 1.16 seconds. What was the vertical distance of his jump, rounded to the nearest tenth of a foot? 100

101 Section 10.7: COMPLEX NUMBERS When you are done with your homework you should be able to Express square roots of negative numbers in terms of i Add and subtract complex numbers Multiply complex numbers Divide complex numbers Simplify powers of i WARM-UP: Rationalize the denominator: a. 5 3 x b. 3 x x THE IMAGINARY UNIT i The imaginary unit is defined as 101

102 THE SQUARE ROOT OF A NEGATIVE NUMBER If b is a positive real number, then Example 1: Write as a multiple of i. a. 100 b. 50 COMPLEX NUMBERS AND IMAGINARY NUMBERS The set of all numbers in the form with real numbers a and b, and i, the imaginary unit, is called the set of. The real number is called the complex number is called an number. Example : Express each number in terms of i and simplify, if possible. a. 7 4 b

103 ADDING AND SUBTRACTING COMPLEX NUMBERS 1. a bi c di =. a bi c di = Example 3: Add or subtract as indicated. Write the result in the form a a. 6 5i 4 3i b. 7 3i 9 10i bi. MULTIPLYING COMPLEX NUMBERS Multiplication of complex numbers is performed the same way as multiplication of, using the property and the FOIL method. After completing the multiplication, we replace any occurrences of with. Example 4: Multiply. a. 5 8i 4i 3 b. 7i 7i c

104 CONJUGATES AND DIVISION The of the complex number a of the complex number a bi is. The bi is. Conjugates are used to complex numbers. The goal of the division procedure is to obtain a real number in the. This real number becomes the denominator of and in. By multiplying the numerator and denominator of the quotient by the of the denominator, you will obtain this real number in the denominator. Example 5: Divide and simplify to the form a bi. a. b. 9 8i 3 4 i d. 6 3 i 4 i c. 5i 3i e. 1 i 1 i 104

105 SIMPLIFYING POWERS OF i 1. Express the given power of i in terms of.. Replace with and simplify. Example 6: Simplify. a. 14 i b. 15 i c. 46 i d. i 6 105

106 Section 11.1: THE SQUARE ROOT PROPERTY AND COMPLETING THE SQUARE; DISTANCE AND MIDPOINT FORMULAS When you are done with your homework you should be able to Solve quadratic equations using the square root property Complete the square of a binomial Solve quadratic equations by completing the square Solve problems using the square root property Find the distance between two points Find the midpoint of a line segment WARM-UP: Solve. a. x 1 4 b. x 5 0 THE SQUARE ROOT PROPERTY If u is an algebraic expression and d is a nonzero real number, then if, then or. Equivalently, if, then or. 106

107 Example 1: Solve. If possible, simplify radicals or rationalize denominators. Express imaginary solutions in the form a bi. a. x 9 d. x 10x 5 1 b. x 10 0 e. 3 x 36 c. 4x

108 COMPLETING THE SQUARE b If x bx is a binomial, then by adding, which is the square of the of, a perfect square trinomial will result. x bx = Example : Find b for each expression. a. x x b. x 1x c. x 5x SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE Consider a quadratic equation in the form ax bx c. 1. If a 1, divide both sides of the equation by.. Isolate x bx. 3. Add to BOTH sides of the equation. 4. Factor and simplify. 5. Apply the square root property. 6. Solve. 7. Check your solution(s) in the equation. 108

109 Example 3: Solve. If possible, simplify radicals or rationalize denominators. Express imaginary solutions in the form a bi. a. x 8x 0 b. x 3x

110 c. 3x 6x d. 4x x

111 A FORMULA FOR COMPOUND INTEREST Suppose that an amount of money,, is invested at interest rate,, compounded annually. In years, the amount,, or balance, in the account is given by the formula Example 4: You invested $4000 in an account whose interest is compounded annually. After 3 years, the amount, or balance, in the account is $4300. Find the annual interest rate. THE PYTHAGOREAN THEOREM The sum of the squares of the of the of a triangle equals the of the of the. If the legs have lengths and, and the hypotenuse has length, then 111

112 Example 5: The doorway into a room is 4 feet wide and 8 feet high. What is the diameter of the largest circular tabletop that can be taken through this doorway diagonally? THE DISTANCE FORMULA The distance,, between the points and in the rectangular coordinate system is 11

113 Example 6: Find the distance between each pair of points. a. 5,1 and 8, b. 3, 6 and 3, 5 6 THE MIDPOINT FORMULA Consider a line segment whose endpoints are and. The coordinates of the segment s midpoints are Example 7: Find the midpoint of the line segment with the given endpoints. a. 10, 4 and,6 b. 7 4, and,

114 Section 11.: THE QUADRATIC FORMULA When you are done with your homework you should be able to Solve quadratic equations using the quadratic formula Use the discriminant to determine the number and type of solutions Determine the most efficient method to use when solving a quadratic equation Write quadratic equations from solutions Use the quadratic formula to solve problems WARM-UP: Solve for x by completing the square and applying the square root property. ax bx c 0 114

115 THE QUADRATIC FORMULA The solutions of a quadratic equation in standard form are given by the quadratic formula: ax bx c 0, with a 0, STEPS FOR USING THE QUADRATIC FORMULA 1. Write the quadratic equation in form ( ).. Determine the numerical values for,, and. 3. Substitute the values of,, and into the quadratic formula and the expression. 4. Check your solution(s) in the equation. Example 1: Solve. If possible, simplify radicals or rationalize denominators. Express imaginary solutions in the form a bi. a. 4x 3x 115

116 b. 3x 4x 6 c. x x 4 3x 3 d. x 5x

117 THE DISCRIMINANT The quantity, which appears under the sign in the formula, is called the. The discriminant determines the and of solutions of quadratic equations. DISCRIMINANT b 4ac KINDS OF SOLUTIONS TO ax bx c 0 GRAPH OF y ax bx c b 4ac 0 b 4ac 0 1 b 4ac 0 NO 117

118 Example : Compute the discriminant. Then determine the number and type of solutions. a. x 4x 3 0 b. 4x 0x 5 c. x x 3 0 DESCRIPTION AND FORM OF THE QUADRATIC EQUATION MOST EFFICIENT SOLUTION METHOD ax bx c 0, and easily factored. ax bx c can be ax c 0 The quadratic equation has no term ( ). u d ; u is a first-degree polynomial. ax bx c 0, and ax bx c cannot factored or the factoring is too difficult. 118

119 THE ZERO-PRODUCT PRINCIPLE IN REVERSE If or, then. Example 3: Write a quadratic equation with the given solution set. a.,6 b. 3, 3 c. i, i Example 4: The hypotenuse of a right triangle is 6 feet long. One leg is feet shorter than the other. Find the lengths of the legs. 119

120 Section 11.3: QUADRATIC FUNCTIONS AND THEIR GRAPHS When you are done with your homework you should be able to Recognize characteristics of parabolas Graph parabolas in the form f x a x h k Graph parabolas in the form f x a x h k Determine a quadratic function s minimum or maximum value Solve problems involving a quadratic function s minimum or maximum value WARM-UP: Graph the following functions by plotting points. a. f x x x f x x x, y b. f x x x f x x x, y 10

121 QUADRATIC FUNCTIONS IN THE FORM f x a x h k The graph of is a whose is the point. The parabola is with respect to the line. If, the parabola opens upwards; if, the parabola opens. f x a x h k 11

122 GRAPHING QUADRATIC FUNCTIONS WITH EQUATIONS IN THE FORM f x a x h k 1. Determine whether the opens or. If the parabola opens upward and if the parabola opens.. Determine the of the parabola. The vertex is. 3. Find any by solving. 4. Find the by computing. 5. Plot the, the, and additional points as necessary. Connect these points with a curve that is shaped like a or an inverted bowl. Example 1: Use the vertex and intercepts to sketch the graph of each quadratic function. Use the graph to identify the function s range. a. f x x 1 1

123 b. f x x 1 THE VERTEX OF A PARABOLA WHHOSE EQUATION IS f x ax bx c The parabola s vertex is. The is and the is found by substituting the into the parabola s equation and the function at this value of. Example : Find the coordinates of the vertex for the parabola defined by the given quadratic function. a. f x 3x 1x 1 b. f x x 7x 4 c. f x 3 x 1 13

124 GRAPHING QUADRATIC FUNCTIONS WITH EQUATIONS IN THE FORM f x ax bx c 1. Determine whether the opens or. If the parabola opens upward and if the parabola opens.. Determine the of the parabola. The vertex is. 3. Find any by solving. 4. Find the by computing. 5. Plot the, the, and additional points as necessary. Connect these points with a curve that is shaped like a or an inverted bowl. Example 3: Use the vertex and intercepts to sketch the graph of each quadratic function. Use the graph to identify the function s range. f x x x 15 a. 14

125 f x 5 4x x b. MINIMUM AND MAXIMUM: QUADRATIC FUNCTIONS Consider the quadratic function f x ax bx c. 1. If, then has a that occurs at. This is.. If, then has a that occurs at. This is. In each case, the value of gives the of the minimum or maximum value. The value of, or, gives that minimum or maximum value. 15

126 Example 4: Among all pairs of numbers whose sum is 0, find a pair whose product is as large as possible. What is the maximum product? Example 5: You have 00 feet of fencing to enclose a rectangular plot that borders on a river. If you do not fence the side along the river, find the length and width of the plot that will maximize the area. What is the largest area that can be enclosed? 16

127 Section 11.4: EQUATIONS QUADRATIC IN FORM When you are done with your homework you should be able to Solve equations that are quadratic in form WARM-UP: Solve. If possible, simplify radicals or rationalize denominators. Express imaginary solutions in the form a bi. a. 5x x 3 b. x x 6 17

128 EQUATIONS WHICH ARE QUADRATIC IN FORM An equation that is in is one that can be expressed as a quadratic equation using an appropriate. In an equation that is quadratic in form, the factor in one term is the of the variable factor in the other variable term. The third term is a. By letting equal the variable factor that reappears squared, a quadratic equation in will result. Solve this quadratic equation for using the methods you learned earlier. Then use your substitution to find the values for the in the equation. Example 1: Solve. If possible, simplify radicals or rationalize denominators. Express imaginary solutions in the form a bi. a. x 13x

129 b. x 4x 5 4 c. x x

130 d. x x e. x 6x

131 Section 1.1: EXPONENTIAL FUNCTIONS When you are done with your homework you should be able to Evaluate exponential functions Graph exponential functions Evaluate functions with base e Use compound interest formulas WARM-UP: Solve. If possible, simplify radicals or rationalize denominators. Express imaginary solutions in the form a bi. x x 6 DEFINITION OF AN EXPONENTIAL FUNCTION The exponential function with base is defined by where is a constant other than ( and ) and is any real number. 131

132 Example 1: Determine if the given function is an exponential function. a. f x 3 x b. 1 g x 4 x Example : Evaluate the exponential function at x, 0, and. a. f x x b. g x x

133 Example 3: Sketch the graph of each exponential function. a. f x 3 x g x b. 3 x How are these two graphs related? 133

134 Example 4: Sketch the graph of each exponential function. a. f x x g x x 1 b. How are these two graphs related? 134

135 CHARACTERISTICS OF EXPONENTIAL FUNCTIONS OF THE FORM x f x b x 1. The domain of f x b consists of all real numbers:. The range x of f x b consists of all real numbers:. x. The graphs of all exponential functions of the form f x b pass through the point because ( ). The is. x 3. If, f x b has a graph that goes to the and is an function. The greater the value of, the steeper the. x 4. If, f x b has a graph that goes to the and is a function. The smaller the value of, the steeper the. x 5. The graph of f x b approaches, but does not touch, the. The line is a asymptote. 135

136 n 1 1 n 1 n The irrational number, approximately, is called the base. The function is called the exponential function FORMULAS FOR COMPOUND INTEREST After years, the balance, in an account with principal and annual interest rate (in decimal form) is given by the following formulas: 1. For compounding interest periods per year:. For continuous compounding: 136

137 Example 5: Find the accumulated value of an investment of $5000 for 10 years at an interest rate of 6.5% is the money is a. compounded semiannually: b. compounded monthly: c. compounded continuously: 137

138 Section 1.: LOGARITHMIC FUNCTIONS When you are done with your homework you should be able to Change from logarithmic to exponential form Change from exponential to logarithmic form Evaluate logarithms Use basic logarithm properties Graph logarithmic functions Find the domain of a logarithmic function Use common logarithms Use natural logarithms WARM-UP: Graph y x. 138

139 DEFINITION OF THE LOGARITHMIC FUNCTION For and,, is equivalent to. The function is the logarithmic function with base. Example 1: Write each equation in its equivalent exponential form: a. log4 x b. y log3 81 Example : Write each equation in its equivalent logarithmic form: y a. e 9 b. 4 b 16 Example 3: Evaluate. a. log5 x 5 log b x 139

140 BASIC LOGARITHMIC PROPERTIES INVOLVING 1 1. log b b the power to which I raise to get is. log 1 b the power to which I raise to get is INVERSE PROPERTIES OF LOGARITHMS For and, 1. log b x b. logb b x Example 4: Evaluate. a. log6 6 c. log9 1 b. log log7 4 d. 7 Example 5: Sketch the graph of each logarithmic function. log3 f x x 140

141 CHARACTERISTICS OF LOGARITHMIC FUNCTIONS OF THE FORM f x log b x 1. The domain of f x log b x consists of all positive real numbers:. The range of f x log b x consists of all real numbers:.. The graphs of all logarithmic functions of the form f x log b x pass through the point because ( ). The is. There is no. 3. If, f x log b x has a graph that goes to the and is an function. 4. If, f x log b x has a graph that goes to the and is a function. 5. The graph of f x log b x approaches, but does not touch, the. The line is a asymptote. 141

142 Example 6: Find the domain. a. f x log x 4 b. f x log 1 x 5 COMMON LOGARITHMS The logarithmic function with base is called the common logarithmic function. The function is usually expressed as. A calculator with a LOG key can be used to evaluate common logarithms. Example 7: Evaluate. a. log1000 b. log 0.01 PROPERTIES OF COMMON LOGARITHMS 1. log1. log10 3. log10 x 4. log 10 x Example 8: Evaluate. a. 3 log10 b. log

143 NATURAL LOGARITHMS The logarithmic function with base is called the natural logarithmic function. The function is usually expressed as. A calculator with a LN key can be used to evaluate common logarithms. PROPERTIES OF NATURAL LOGARITHMS 1. ln1. ln e x 3. ln e 4. ln x e Example 9: Evaluate. 1 a. ln b. e 6 ln300 e Example 10: Find the domain of f x ln x

144 Section 1.3: PROPERTIES OF LOGARITHMS When you are done with your 1.3 homework you should be able to Use the product rule Use the quotient rule Use the power rule Expand logarithmic expressions Condense logarithmic expressions Use the change-of-base property WARM-UP: Simplify. x a. 5 5 x b. 3 x x THE PRODUCT RULE Let,, and be positive real numbers with. The logarithm of a product is the of the. Example 1: Expand each logarithmic expression. a. log6 6x b. ln x x 144

145 THE QUOTIENT RULE Let,, and be positive real numbers with. The logarithm of a quotient is the of the. Example : Expand each logarithmic expression. a. 1 x log b. log4 x THE POWER RULE Let,, and be positive real numbers with, and let be any real number. The logarithm of a quotient is the of the. Example 3: Expand each logarithmic expression. a. log x b. log5 x 145

146 PROPERTIES FOR EXPANDING LOGARITHMIC EXPRESSIONS For and : 1. = log M log. = log M log b b b b N N 3. = p log b M Example 4: Expand each logarithmic expression. a. 3 log x 3 y log x 1y b. 4 5 PROPERTIES FOR CONDENSING LOGARITHMIC EXPRESSIONS For and : 1. = log b MN. = log b M N 3. = log M p b 146