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1Click to go to website: www.njctl.org New Jersey Center for Teaching and LearningProgressive Mathematics InitiativeThis material is made freely available at and is intended for the non-commercial use of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJCTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others.Click to go to website:
2Linear, Exponential and Logarithmic Functions Algebra IILinear, Exponential and Logarithmic Functions
3Logarithmic Functions click on the topic to goto that sectionTable of ContentsLinear FunctionsExponential FunctionsLogarithmic FunctionsProperties of LogsSolving Logarithmic Equationse and lnGrowth and Decay
5Linear FunctionsGoals and ObjectivesStudents will be able to analyze linear functions using x and y intercepts, slope and different forms of equations.
6Linear FunctionsWhy do we need this?Being able to work with and analyze lines is very important in the fields of Mathematics and Science. Many different aspects of life come together in linear relationships. For example, height and shoe size, trends in economics or time and money. Quickly, even these situations become non-linear, but we can still model some information using lines.
7We will begin with a review of linear functions: a) x and y intercepts b) Slope of a linec) Different forms of lines: i) Slope-intercept form of a line ii) Standard form of a line iii) Point-slope form of a lined) Horizontal and vertical linese) Parallel and perpendicular linesf) Writing equations of lines in all three forms
8Definitions: x and y intercepts Teacherx-int = (-6, 0)y-int = (0, 8)Linear FunctionsDefinitions: x and y interceptsGraphically, the x-intercept is where the graph crosses the x-axis. To find it algebraically, you set y = 0 and solve for x.Graphically, the y-intercept is wherethe graph crosses the y-axis. To find italgebraically, you set x = 0 and solve for y.yx246810-2-4-6-8-10Find the x and y intercepts on the graph to the right. Write answers as coordinates.x-int =y-int =
9Teacherx-int = (5, 0)y-int = (0, 3)Linear FunctionsFind the x and y intercepts on the graph to the right. Write answers as coordinates.yx246810-2-4-6-8-10x-int =y-int =
10TeacherSlopeHave students then figure out the slope of these three lines.Dark Blue = -2Green = 0Red = -1/2Purple = 4/3yx246810-2-4-6-8-10Linear FunctionsAn infinite number of lines can pass through the same location on the y-axis...they all have the same y-intercept.Examples of lines with a y-intercept of ____ are shown on this graph. What is the difference between them (other than their color)?
11You can find slope two ways: Linear FunctionsDef: The Slope of a line is the ratio of its rise over its run. For notation, we use "m."TeacherStress the difference and advantages to both methods.You can find slope two ways:yx246810-2-4-6-8-10runriseAlgebraically:Graphically: count
12y m = 4/3 rise run x Find the slope of this line. m = Linear Functions 246810-2-4-6-8-10riserunTeacherm = 4/3Find the slope of this line.m =
13rise run 1 Find the slope of the line to the right. m = -3/4 Linear FunctionsTeacherm = -3/4Make the rise and run bigger or smaller to emphasize that there are many places to find slope.1Find the slope of the line to the right.yx246810-2-4-6-8-10(Move the rise and the run to fit the slope. Make them bigger or smaller to get an accurate slope.)runrise
14rise run 2 Find the slope of the line to the right: m = 2/3 Linear FunctionsTeacherm = 2/3Make the rise and run bigger or smaller to emphasize that there are many places to find slope.yx246810-2-4-6-8-102Find the slope of the line to the right:(Move the rise and the run to fit the slope. Make them bigger or smaller to get an accurate slope.)runrise
15Find the slope of the line going through the following points.
16Find the slope of a line going through the following points: Linear FunctionsTeacherm = -8/73Find the slope of a line going through the following points:(-3, 5) and (4, -3)
17Find the slope of the line going through the following points. Teacherm = 3/5Linear Functions4Find the slope of the line going through the following points.(0, 7) and (-5, 4)
18Roofs Distance Height And many more... TeacherGet ideas from students where slope is used in the "real world."Linear FunctionsSlope formula can be used to find the constant of change in "real world" problems.Mountain HighwaysRoofsGrowthDistanceHeightAnd many more...
19Linear FunctionsWhen traveling on the highway, drivers will set the cruise control and travel at a constant speed. This means that the distance traveled is a constant increase.The graph below represents such a trip. The car passed mile- marker 60 at 1 hour and mile-marker 180 at 3 hours. Find the slope of the line and what it represents.Time(hours)Distance(miles)(1,60)(3,180)
20Teacherm = 50mphLinear Functions5If a car passes mile-marker 100 in 2 hours and mile-marker 200 after 4 hours, how many miles per hour is the car traveling?
21Teacherm = 7.5 meters per secondLinear Functions6How many meters per second is a person running if they are at 10 meters in 3 seconds and 100 meters in 15 seconds?
22Standard Form Point-Slope Form y = mx + b y - y1 = m(x - x1) Linear FunctionsWe are going to look at three different forms of the equations of lines. Each has its advantages and disadvantages in their uses.Slope-Intercept FormStandard FormPoint-Slope Formy = mx + bAx + By = Cy - y1 = m(x - x1)Advantage:Easy to find slope, y- intercept and graph the line from the equation.Disadvantage:Must be solved for y.Easy to find intercepts and graph.Must manipulate it algebraically to find slope.Can find equation or graph from slope and any point.Cumbersome to put in another form.
23Determine the equation of the line from the given graph. 1) Determine the y-intercept.Remember, graphically, the y-intercept iswhere the graph crosses the y-axis.clickb = -32) Graphically find the slopefrom any two points.click3) Write the equation of the lineusing the slope-intercept form.y = mx + by = 2x - 3click
24Which equation does this line represent? TeacherD y = x - 6Ask students which form it is in:Slope-interceptLinear Functions7Which equation does this line represent?Ay = -6x - 6By = -x - 6Cy = 6x + 6Dy = x - 6
25Which graph represents the equation y = 3x - 2? Linear FunctionsTeacherD line DABCD8Which graph represents the equation y = 3x - 2?ALine ABLine BCLine CDLine D
26What equation does line A represent? 9What equation does line A represent?TeacherBABCDAy = 2x + 3By = -2x + 3Cy = 0.5x + 3Dy = -0.5x + 3
27What equation does line B represent? 10What equation does line B represent?TeacherDAy = 2x + 3ABCDBy = -2x + 3Cy = 0.5x + 3Dy = -0.5x + 3
28Consider the equation 4x - 3y = 6. Linear FunctionsConsider the equation 4x - 3y = 6.Which form is it in? Graph it using the advantages of the form.TeacherIt is in Standard Form. Graph it using the x and y intercepts.x-int: (3/2, 0)y-int: (0, -2)
29Graph the equation 5x + 6y = -30 using the most appropriate method based on the form. TeacherIt is in Standard Form. Graph it using the x and y intercepts.x-int: (-6, 0)y-int: (0, -5)
30Graph the equation 3x - 5y = -10 using the most appropriate method based on the form. TeacherIt is in Standard Form. Graph it using the x and y intercepts.x-int: (-10/3, 0)y-int: (0, 2)
31Write the equation in standard form. Multiply both sides of the equation by the LCD (6).Rearrange the equation so that the x and y terms are on the same side together. Make sure that x is always positive.
32Multiply each term by LCD: 15 Linear Functions11What is the Standard Form of:A3x + 5y = 15B9x + 15y = 35TeacherB: 9x + 15y = 35Multiply each term by LCD: 15C15x - 9y = 35D5x - 3y = 15
33Which form is this equation in? y - 3 = 4(x + 2) 12Which form is this equation in?y - 3 = 4(x + 2)Point-Slope FormTeacherAStandard FormBSlope-Intercept FormCPoint-Slope Form
3413 Find y when x = 0. y - 3 = 4(x + 2) Point-Slope Form y is 11 Linear Functions13Find y when x = 0.y - 3 = 4(x + 2)TeacherPoint-Slope Formy is 11
35Put the following information for a line in Point-Slope form: TeacherC y - 5 = -3(x + 2)Linear Functions14Put the following information for a line in Point-Slope form:m = -3 going through (-2, 5).Ay - 3 = -2(x + 5)By - 2 = 5(x - 3)Cy - 5 = -3(x + 2)Dy + 2 = -3(x - 5)
36Put the following information for a line in Point-Slope form: Linear Functions15Put the following information for a line in Point-Slope form:m = 2 going through (1, 4).TeacherC y - 4 = 2(x - 1)Ay + 4 = 2(x + 1)By + 4 = 2(x - 1)Cy - 4 = 2(x - 1)Dy - 4 = 2(x + 1)
37Put the following information for a line in Point-Slope form: Linear Functions16Put the following information for a line in Point-Slope form:m = -2/3 going through (-4, 3).TeacherD y - 3 = -2/3(x + 4)Ay + 3 = -2/3(x + 4)By - 3 = -2/3(x - 4)Cy + 3 = -2/3(x - 4)Dy - 3 = -2/3(x + 4)
38Horizontal and Vertical Lines Linear FunctionsHorizontal and Vertical LinesExample of Horizontal line y=2Example of a Vertical Line x=3Horizontal lines have a slope of 0.*Notice that a horizontal line will "cut" the y axis and has the equation of y = m.Vertical lines have an undefined slope.*Notice that a vertical line will "cut" the x axis and has the equation of x = n.
39Which equation does the line represent? Linear FunctionsTeacherC y = 4Easy way to remember horizontal and vertical equations:Horizontal lines "cut" the y-axis at the number indicated. Therefore, it is y = n.Vertical lines "cut" the x-axis at the number indicated. Therefore, it is x = m.17Which equation does the line represent?Ay = 4xBy = x + 4Cy = 4Dy = x
40TeacherB HorizontalLinear Functions18Describe the slope of the following line: y = 4 (Think about which axis it "cuts.")AVerticalBHorizontalCNeitherDCannot be determined
41Describe the slope of the following line: 2x - 3 = 4. Linear FunctionsTeacherA Vertical19Describe the slope of the following line: 2x - 3 = 4.AVerticalBHorizontalCNeitherDCannot be determined
42Parallel and Perpendicular Lines Parallel lines have the same slope:Perpendicular lines have opposite , reciprocal slopes.Perpendicular lines have opposite , reciprocal slopes.h(x) = x + 6q(x) = x + 2r(x) = x - 1s(x) = x - 5h(x) = -3x - 11g(x) = 1/3x - 2
43Drag the equation to complete the statement. Linear FunctionsDrag the equation to complete the statement.TeacherHave students come to the board and drag.A) y = -1/4x - 3B) 1/5y = x - 2C) y = 4x - 1D) y = -1/5x + 9E) 6x + y = 10F) y = 1/5xG) y = 1/6x - 6A) y = 4x - 2 is perpendicular to _____________.B) y = -1/5x + 1 is perpendicular to _____________.C) y - 2 = -1/4(x - 3) is perpendicular to _____________.D) 5x - y = 8 is perpendicular to _____________.E) y = 1/6x is perpendicular to _____________.F) y - 9 = -5(x - .4) is perpendicular to _____________.G) y = -6(x + 2) is perpendicular to _____________.
44How to turn an equation from point-slope to slope-intercept form. standard form to slope-intercept form.Rearrange the equation so that the y term is isolated on the right hand side.DistributeRearrange the equation so that the y term is isolated on the right hand side.Make sure that y is always positive.
45Are the lines 2x - 3y = 7 and 4x - 6y = 11 parallel? Turn the equations into slope-intercept form.Both slopes are 2/3. Yes the lines are parallel.
46Both slopes are 2/5. Yes the lines are parallel. Are the lines 2x + 5y = 12 and 8x + 20y = 16 parallel?TeacherBoth slopes are 2/5. Yes the lines are parallel.Turn the equations into slope-intercept form.
47Which line is perpendicular to y = -3x + 2? Linear FunctionsTeacherB20Which line is perpendicular to y = -3x + 2?Ay = -3x + 1BCy = 5Dx = 2
48Which line is perpendicular to y = 0? Linear FunctionsTeacherD x = 221Which line is perpendicular to y = 0?Ay = -3x + 1By = xCy = 5Dx = 2
49TeacherSlope-Intercept Form: Easiest to use if you have slope and y-intercept. If not, you will have to solve for b.Standard Form: Not very useful to find equations, but easy to find intercepts and graph with.Point-Slope Form: Easiest form to use in general. The downside is most questions will ask for another form.Linear FunctionsWriting Equations of LinesRemember the three different forms of the equations of lines. Brainstorm when each form would be easiest to use when having to write an equation of the line.Slope-Intercept FormStandard FormPoint-Slope Formy = mx + bAx + By = Cy - y1 = m(x - x1)
50Writing an equation in Slope- Intercept Form: Linear FunctionsWriting an equation in Slope- Intercept Form:From slope and one or two points:1. Find the slope. Use either given information or the formula:2. Plug the coordinates from one point into x and y respectfully.3. Solve for b.4. Write as y = mx + b.From slope and y-intercept:1. Just plug into slope into m and y-intercept into b.2. Write as y = mx + b
51Write an equation in slope-intercept form through the given points. Find the slope.Plug the coordinates from one point into x and y respectfully.
52Linear FunctionsTeacherWrite the equation of the line with slope of 1/2 and through the point (2,5). Leave your answer in slope-intercept form.
53Linear FunctionsTeacherWrite the equation of the line through (-3,-2) and perpendicular to y = -4/5x + 1 in slope-intercept form.
54Linear Functions22Write the equation of the line going through the points (1, 2) and (3, -4). Your answer should be in slope-intercept form.TeacherAy = -3x + 5y = -3x - 3y = -3x + 2BCD
55Linear FunctionsTeacher23What is the equation of the line with a slope parallel to 3x + y = 2 going through the point (4, -2)Ay = -3x + 10By = -3x - 14Cy = 3x + 10Dy = -3x + 14
56Writing an equation in Point-Slope Form: Linear FunctionsWriting an equation in Point-Slope Form:From slope and one or two points:1. Find the slope. Use either given information or the formula:2. Plug the coordinates from one point into x1 and y1 respectfully.3. Write as y - y1 = m(x - x1).
57Write an equation in point-slope form through the given two points. Find the slope.Use either point.
58Write the equation of the line through Linear FunctionsWrite the equation of the line through(5,6) and (7,1) using Point-Slope form.Teacheror
59Linear FunctionsWrite the equation of the line with slope of 1/2 and through the point (2,5) in Point-Slope form.Teacher
60Linear FunctionsTeacher24Which of the following is the Point-Slope form of the line going through (-2, 4) and (6, -2)?ABCD
6125 A y - 3 = 2(x - 3) B y = 2(x - 3) C y - 3 = 2x D y = 3 TeacherLinear Functions25Which of the following is the Point-Slope form of the line going through (0, 3) having the slope of 2?Ay - 3 = 2(x - 3)By = 2(x - 3)Cy - 3 = 2xDy = 3
62Teacher4x - y = 11Have a class discussion about the usefulness of this form and how easy this was to do.Linear FunctionsStandard FormWrite the equation of the line with m= 4, through (3,1) and (4,5) in Standard Form.Put the equation into point-slope form. Then turn it into standard form.DistributeDistributeRearrange the equation so that the x and y terms are on the same side together.Rearrange the equation so that the x and y terms are on the same side together.Make sure that x is always positive.Make sure that x is always positive.
63How to turn an equation from point-slope to standard form. Teacherx + y = -6Have a class discussion about the usefulness of this form and how easy this was to do.Write the equation of the line through (-1,-5) and (-4,-2) in Standard Form.
64Write the equation of the line through (-7, -2) and (1, 6) in Standard Form. Teacherx - y = 5Have a class discussion about the usefulness of this form and how easy this was to do.
65Write the equation of the line with Linear FunctionsWrite the equation of the line withx-intercept of 5 and y-intercept of 10. Which form would be easiest to use?Teachery = -2x + 10ory - 10 = -2xy = -2(x - 5)There is nothing that states what form you must leave answer in, so many answers are correct.
66Write the equation of the line through (4,1) and Linear FunctionsWrite the equation of the line through (4,1) andparallel to the line y = 3x Which form is easiest to use in this case?Teachery - 1 = 3(x - 4)y = 3x -113x - y = 11
67TeacherThis is an opinion question that fuels discussion about different forms and their purpose.Linear Functions26Given a point of (-3, 3) and a slope of -2, what is the easiest form to write the equation in?ASlope-intercept FormBPoint-Slope FormCStandard Form
68Linear Functions27Given two points on a line, (3, 0) and (0, 5), what is the easiest form to write the equation in?TeacherThis is an opinion question that fuels discussion about different forms and their purpose.ASlope-intercept FormBPoint-Slope FromCStandard Form
69Linear FunctionsTeacherAll of the answers are equations of the line, but only A and B are in Point-Slope form.28Which of the following is/are the equation(s) of the line through (1, 3) and (2, 5) in Point-Slope form?Ay - 5 = 2(x - 2)By - 3 = 2(x - 1)Cy = 2x + 1D2x - y = -1
71Exponential Functions Goals and ObjectivesStudents will be able to recognize and graph exponential functions.
72Exponential Functions Why do we need this?Most situations that people study do not have linear relationships. Population growth is now heavily studied around the world. Is this a linear function? Why do we study population growth?
73Exponential Functions We have looked at linear growth, where the amount of change is constant. This is a graph of the amount of money your parents give you based on the number of A's that you receive on your report card.XY110220330440Can we easily predict what you would get if you have 7 A's?$$Number of A's
74Exponential Functions TeacherCapitalize on this problem to emphasize the difference between linear and exponential relationships. Which equation is better for your pocketbook? But is it realistic?The equation for the previous function is y = 10x.What if you asked your parents to reduce the amount per A to $3, but then asked them to use the following function:y = 3xGraph it!!What would you get for 7 A's?
75y = 3x Can you find the amount for 7 A's? Exponential FunctionsHere is the graph of your function! This is an example of exponential growth.y = 3xCan you find the amount for 7 A's?Would that provide some motivation?
76X Y 80 1 40 2 20 3 10 4 5 Now, let's look an opposite problem: TeacherDraw the curve on the graph. Ask students if the curve will ever go below the x-axis and why?Exponential FunctionsNow, let's look an opposite problem:Suppose you are given 80 M&M's and each day you eat half. What does this curve look like? This is an example of exponential decay.XY8014022031045
77Here is a graph of the M&M problem. Teacher1. It crosses at 80. That is the amount we start with.2. Day 2: 20, Day 3:10, Day 4: 5,Day 5: Either 2, 2.5, or 3. Have a discussion about how you would deal with the issue of "half of 5" M&M's.3. Bring up asymptotes and discuss the possibility that the M&M's may never be gone if we take half of one, then half of the half, etc... Relate it to asymptotes.Exponential FunctionsNumber of M&M'sDaysHere is a graph of the M&M problem.1. Where does the graph cross the y-axis?2. How many M&M's do you have on day 2, 3, 4 and 5?3. When are all of the M&M's gone?
78Exponential Functions Now we are going to identify an Exponential Function both graphically and algebraically.
79Stress and explore domain and range. Graphically TeacherStress and explore domain and range.Exponential FunctionsGraphicallyThe exponential function has a curved shape to it. Y-values in an exponential function will either get bigger or smaller very, very quickly.Domain: (x values)Range: (y values)Exponential GrowthExponential DecayWhy does the domain run from negative infinity to positive infinity?Why does the range run from 0 to positive infinity?
80Which of the following are graphs of exponential growth? Exponential Functions29Which of the following are graphs of exponential growth?(You can choose more than one.)TeacherA and FABCDEFGH
81TeacherDExponential Functions30Which of the following are graphs of exponential decay? (You can choose more than one.)ABCDEFGH
82The general form of an exponential function is TeacherStudents will struggle with b. Make sure you stress the difference between b > 1 and 0 < b < 1 related to growth and decay. Discuss why 1 is not included.Exponential FunctionsThe general form of an exponential function iswhere x is the variable and a, b, and c are constants.b is the growth rate.If b > 1 then it is exponential growthIf 0 < b < 1 then it is exponential decayy = c is the horizontal asymptote(0, a + c) is the y-interceptremember, to find the y-intercept, set x = 0.
83Does the following exponential equation represent growth or decay? Exponential FunctionsTeacherA Growth31Does the following exponential equation represent growth or decay?AGrowthBDecay
84TeacherC y = 4Exponential Functions32For the same function, what is the equation of the horizontal asymptote?Ay = 2By = 3Cy = 4Dy = 5
85Now, find the y-intercept: TeacherC (0, 7)Exponential Functions33Now, find the y-intercept:A(0, 3)B(0, 4)C(0, 7)D(0, 9)
86Exponential Functions 34Now consider this new exponential equation: Does it represent growth or decay?AGrowth BDecayTeacherB Decay
87Which of the following is the equation of the horizontal asymptote? TeacherC y = 3Exponential Functions35Which of the following is the equation of the horizontal asymptote?Ay = 0.2By = 1Cy = 3Dy = 4
88Find the y-intercept in the same function: Exponential Functions36Find the y-intercept in the same function:A(0, 0.2)TeacherD (0, 4)B(0, 1)C(0, 3)D(0, 4)
89B Decay. Rewrite the function with a positive exponent: TeacherB Decay. Rewrite the function with a positive exponent:Have a discussion on what is different about this function.Exponential Functions37Does the following exponential function represent growth or decay?AGrowthBDecay
90Find the horizontal asymptote for the following function: Exponential FunctionsTeacherA y = 038Find the horizontal asymptote for the following function:Ay = 0By = 1Cy = 3Dy = 4
91For the same function, what is the y-intercept? Exponential FunctionsTeacherD (0, 4)39For the same function, what is the y-intercept?A(0, 0)B(0, 1)C(0, 3)D(0, 4)
921) Identify horizontal asymptote (y = c) TeacherExponential FunctionsTo sketch the graph of an exponential function, use the values for a, b and c.1) Identify horizontal asymptote (y = c)2) Determine if graph is decay or growth3) Graph y-intercept (0,a+c)4) Sketch graphTry it!Note: A horizontal asymptote is the horizontal line at y = c that the exponential function cannot pass.
98Logarithmic Functions Goals and ObjectivesStudents will be able to rewrite exponential equations as logarithmic equations, rewrite logarithmic equations as exponential equations, manipulate logarithmic expressions, solve exponential equations and logarithmic equations.
99Logarithmic Functions Why do we need this?Logarithms are used to simplify calculations. They make certain exponential equations much easier to solve and allow us to study how exponents affect functions.
100A logarithmic function is the inverse of an exponential function. Logarithmic FunctionsA logarithmic function is the inverse of an exponential function.+=Graphically, the inverse of any function is the reflection of that function over the line y = x. Essentially, the x and y coordinates become switched.
101Logarithmic Functions Logarithmic functions switch the domain and range of exponential functions, just like points are switched in inverse functions.Domain:Range:*Notice that 0 and any negative are not in the domain of a logarithm. Therefore, you cannot take the log of 0 or a negative.
102baseexponent = answer logbaseanswer = exponent Logarithmic FunctionsLogarithms are a way to study the behavior of an exponent in an exponential function. Here is how to rewrite an exponential equation as a logarithmic equation (and visa versa):baseexponent = answerlogbaseanswer = exponent*A log that appears to have no base is automatically considered base 10. For example:
103baseexponent = answer logbaseanswer = exponent Logarithmic Functionsbaseexponent = answerlogbaseanswer = exponentTeacherRewrite the following in logarithmic form:
104baseexponent = answer logbaseanswer = exponent Logarithmic Functionsbaseexponent = answerlogbaseanswer = exponentTeacherRewrite the following in exponential form.
105Logarithmic Functions TeacherA42Which of the following is the correct logarithmic form of:ACBD
106Logarithmic Functions TeacherD43Which of the following is the correct logarithmic form of:ACBD
10744 Which of the following is the correct exponential form of: B A B C TeacherBLogarithmic Functions44Which of the following is the correct exponential form of:ABCD
108Which of the following is the correct exponential form of: Logarithmic FunctionsTeacherC45Which of the following is the correct exponential form of:ACBD
109What is the exponential form of: Logarithmic FunctionsTeacherA46What is the exponential form of:ABCD
110Logarithmic Functions Convert each of the following to exponential form in order to either simplify the logarithmic expression or to find the value of the variable.Teacher
116Use this as a reminder of negative exponents and what they mean. 52 Logarithmic FunctionsTeacher-4Use this as a reminder of negative exponents and what they mean.52Evaluate:
117Use this as a reminder of negative exponents and what they mean. Teacher-3Use this as a reminder of negative exponents and what they mean.Logarithmic Functions53Evaluate:
118TeacherNo solution. You cannot take the log of a negative number.Logarithmic Functions54Evaluate:
119TeacherYou need a calculator to do this one. We will cover it in the next few slides.Logarithmic Functions55Evaluate:
120TeacherLogarithmic FunctionsTake out a calculator and find "log". What base is it?Use the calculator to find the following values. Round off to three decimal places.
121Teacher5.129Make sure that students close parentheses and put them in the right place.Logarithmic FunctionsEvaluate:Since the log button on your calculator is only base 10, we need a change of base to get the value of something other than base 10.
122Logarithmic Functions Teacher1.827Make sure that students close parentheses and put them in the right place.56Evaluate:
123Logarithmic Functions Teacher1.011Make sure that students close parentheses and put them in the right place.57Evaluate:
12458 Evaluate: -1.679 Logarithmic Functions Teacher-1.679Students will question you about the negative, as there are not logs of negative numbers. Stress that this is returning a negative VALUE, and is not the log of a negative number.58Evaluate:
12559 Evaluate: -0.900 Logarithmic Functions TeacherStudents will question you about the negative, as there are not logs of negative numbers. Stress that this is returning a negative VALUE, and is not the log of a negative number.-0.900Logarithmic Functions59Evaluate:
126Algebraically, how would you solve: Logarithmic FunctionsThe change of base formula and converting to logarithms will now allow you to solve more complicated exponential equations.TeacherThis can be done with a graph, but logarithms make it easier. Put it in logarithmic form and use a calculator to solve.Algebraically, how would you solve:
127Solve the following equation: Logarithmic FunctionsTeacherSolve the following equation:
128Solve the following equation. Logarithmic FunctionsTeacher60Solve the following equation.
154Solving Logarithmic Equations To solve a logarithmic equation, it needs to be put into one of the following forms:*After the equation is in this form, you may need to convert to exponential form.*After the equation is in this form, a and c must be equal. Therefore, you may remove the logarithms and solve.
155Solving Logarithmic Equations TeacherThis equation can be contracted on one side to get one logarithm.Before we solve, should we put this equation into a logarithm on one side or a logarithm on both sides?
156Solve: Solving Logarithmic Equations Teacher The solution r = -1 is actually an extraneous solution as you cannot take a log of x ≤ 0.Solving Logarithmic EquationsSolve:
157Solving Logarithmic Equations TeacherStress that students must check for extraneous solutions when solving logarithmic equations.STOP!Extraneous Solutions: Remember you cannot take a log of x ≤ 0. ALWAYS check to see if your solution(s) work.
158Solve the following equation: Solving Logarithmic EquationsTeacher74Solve the following equation:
159Solve the following equation: TeacherSolving Logarithmic Equations75Solve the following equation:
160Solve the following equation: TeacherSolving Logarithmic Equations76Solve the following equation:
161Solve the following equation: Solving Logarithmic Equations77Solve the following equation:Teacher,
162Solve the following equation: Solving Logarithmic EquationsTeacher78Solve the following equation:
163Solve the following equation: Solving Logarithmic EquationsTeacher79Solve the following equation:
164How can we use these concepts to solve: Solving Logarithmic EquationsTeacherIt is actually legal to take the log of both sides. You can then apply the properties of logarithms to solve.How can we use these concepts to solve:
165Try: Solving Logarithmic Equations TeacherIt is actually legal to take the log of both sides. You can then apply the properties of logarithms to solve.Try:
170Basic Properties of Natural Logs Properties of Natural LogarithmsBasic Properties of Natural Logs
171Formally, e is defined to be: Simply, as a number: e and lnThe letter e, shows up quite often when dealing with exponential functions. It is a number that models things like the growth of a bacteria colony, the spread of an oil spill and even calculating compound interest.Formally, e is defined to be:Simply, as a number:*e is similar to pi in sense that it will never repeat and never ends...
172*Find ln on your calculator. This is loge. e and lnAs with exponential functions, we can find the inverse of a function with base e. This is called The Natural Log and is noted:*Find ln on your calculator. This is loge.
173e and lnThe graphs of e and ln are similar to our other functions. The domain and range also remain the same.
174e and lne and natural logs have all of the same properties that other exponentials and logarithms have.For example:becausebecause
175Write the following in the equivalent exponential or log form. Teachere and lnWrite the following in the equivalent exponential or log form.
176Put the following into a single logarithm: e and lnTeacherPut the following into a single logarithm:
177Expand the following logarithms: Teachere and lnExpand the following logarithms:
178Put into a single logarithm: TeacherDe and ln83Put into a single logarithm:ABCD
179Expand the following logarithm: TeacherBe and ln84Expand the following logarithm:ABCD
180Expand the following logarithm: e and lnTeacherB85Expand the following logarithm:ABCD
181Put into a single logarithm: TeacherAe and ln86Put into a single logarithm:ACBD
182Solve the following equations: e and lnTeacher*Sometimes you will be asked to leave answers as exact numbers.Solve the following equations:
183Solve the following equations: Teacher*Only 60.6 orare solutions because you cannot take the log of a negative.e and lnSolve the following equations:
191Growth and DecayGoals and ObjectivesStudents will be able to model growth and decay problems with exponential equations.
192Growth and DecayWhy do we need this?What will the population of the world be in 2050 at the current rate of growth? How long will it take the radioactive material near the nuclear reactors in Japan to dissipate to harmless levels after the destructive tsunami? These are important questions that we need to answer to plan for the future! Growth and decay can be modeled and analyzed with exponential and logarithmic functions.
193Growth and DecayAnd...these functions will also model problems dealing with something we ALL need to learn to work with...Money!
194 P = the principal (amount deposited) Teacher$ in 2 years.Growth and DecayThis formula represents the amount (A) of money in a savings account if the interest is continually compounded. P = the principal (amount deposited) r = the annual interest rate (in decimal form) t = time in yearsIf $500 is invested at 4% for 2 years, what will account balance be?
195It will take about 17.3 years for the money to double. Growth and DecayTeacherIt will take about 17.3 years for the money to double.If $500 is invested at 4%, compounded continually, how long until the account balance is doubled?
196Growth and Decay93If $1000 is invested at 4% for 3 years, compounded continually, what is the account balance?Teacher
197Growth and DecayTeacher19.8 years94If $1000 is invested at 3.5%, how long until the account balance is doubled?
198Account Balance with Compound Interest: Growth and DecayTeacherSpend time discussing parts of each type of interest.There are other types of interest that you may encounter...I = interestP = principal (deposit)r = interest rate (decimal)t = time in yearsn = number of times compoundedper unit of tA=account balance after theinterest is includedSimple Interest:Account Balance with Simple InterestSiSsSsssAccount Balance with Compound Interest:
199TeacherI = $480Growth and DecayCalculate the Simple interest if you were to buy a car from your parents for $4000 and pay for it over 4 years at 3% interest.
200Ask students which is the better deal. Growth and DecayTeacherI = $1600Ask students which is the better deal.Calculate the simple interest if you were to purchase a car from a used car dealer for 4000 at 10% interest over 4 years.
201TeacherGrowth and DecayUsing the formula for compound interest, calculate the amount in your account if you invest $1300, over 15 years, at 4.6% interest compounded monthly.
202Growth and DecayHow long would it take for an investment of $10,000 to increase to a total of $25,000 compounded quarterly at a rate of 3.5%?Teacher
203Growth and Decay95How many years did you take to pay off a $5000 car at 7% simple interest if you paid a total of $6200 for the car?Teacher
204TeacherGrowth and Decay96What was the rate of your investment if you invested $5000 over 10 years, compounded continuously and you made $2200 in interest?
205TeacherGrowth and Decay97How much was originally invested if you have $63, in an account generating 4% interest (compounded monthly) over 15 years?
206r = rate or growth or decay(decimal) t = time TeacherDiscuss the different uses of the same formulas.Growth and DecayThe same formulas can be used to model growth or decay in other situations.P = Population (Initial)r = rate or growth or decay(decimal)t = time
207Example: A bacteria constantly grows at a rate of Growth and DecayTeacherIt will take the colony 23 hours to increase its population from 100 to 1000.Example: A bacteria constantly grows at a rate of10% per hour, if initially there were 100 how long tillthere were 1000?
208TeacherThe original value of the car was about $30,345.Growth and DecayA new car depreciates in value at a rate of 8% per year. If a 5 year old car is worth $20,000,how much was it originally worth? How will we write the rate?
209A certain radioactive material has a half-life of 20 years. Growth and DecayTeacherIn 7 years, about 78.5g remain.A certain radioactive material has a half-life of 20 years.If 100g were present to start, how much will remain in 7 years?Use half-life of 20 years to find r.
210TeacherIt would take ≈ 2.7 hoursGrowth and Decay98If an oil spill widens continually at a rate of 15% per hour, how long will it take to go from 2 miles wide to 3 miles wide?
211TeacherGrowth and Decay99If you need your money to double in 8 years, what must the interest rate be if is compounded quarterly?
212Growth and DecayTeacher100NASA calculates that a communications satellite's orbit is decaying exponentially at a rate of 12% per day. If the satellite is 20,000 miles above the Earth, how long until it is visible to the naked eye at 50 miles high, assuming it doesn't burn up on reentry?
213Growth and DecayTeacher101If the half-life of an element is 50 years, at what rate does it decay?