# New Jersey Center for Teaching and Learning

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Linear, Exponential and Logarithmic Functions
Algebra II Linear, Exponential and Logarithmic Functions

Logarithmic Functions
click on the topic to go to that section Table of Contents Linear Functions Exponential Functions Logarithmic Functions Properties of Logs Solving Logarithmic Equations e and ln Growth and Decay

Linear Functions Goals and Objectives Students will be able to analyze linear functions  using x and y intercepts, slope and different  forms of equations.

Linear Functions Why 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.

We will begin with a review of linear functions: a) x and y intercepts
b) Slope of a line c) Different forms of lines:  i) Slope-intercept form of a line  ii) Standard form of a line  iii) Point-slope form of a line d) Horizontal and vertical lines e) Parallel and perpendicular lines f) Writing equations of lines in all three forms

Definitions: x and y intercepts
Teacher x-int = (-6, 0) y-int = (0, 8) Linear Functions Definitions: x and y intercepts Graphically, 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 where the graph crosses the y-axis. To find it algebraically, you set x = 0 and solve for y. y x 2 4 6 8 10 -2 -4 -6 -8 -10 Find the x and y intercepts on the graph to the right. Write answers as coordinates. x-int = y-int =

Teacher x-int = (5, 0) y-int = (0, 3) Linear Functions Find the x and y intercepts on the graph to the right. Write answers as coordinates. y x 2 4 6 8 10 -2 -4 -6 -8 -10 x-int = y-int =

Teacher Slope Have students then figure out the slope of these three lines. Dark Blue = -2 Green = 0 Red = -1/2 Purple = 4/3 y x 2 4 6 8 10 -2 -4 -6 -8 -10 Linear Functions An 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)?

You can find slope two ways:
Linear Functions Def: The Slope of a line is the ratio of its rise over its run. For notation, we use "m." Teacher Stress the difference and advantages  to both methods. You can find slope two ways: y x 2 4 6 8 10 -2 -4 -6 -8 -10 run rise Algebraically: Graphically: count

y m = 4/3 rise run x Find the slope of this line. m = Linear Functions
2 4 6 8 10 -2 -4 -6 -8 -10 rise run Teacher m = 4/3 Find the slope of this line. m =

rise run 1 Find the slope of the line to the right. m = -3/4
Linear Functions Teacher m = -3/4 Make the rise and run bigger or  smaller to emphasize that there  are many places to find slope. 1 Find the slope of the line to the right. y x 2 4 6 8 10 -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.) run rise

rise run 2 Find the slope of the line to the right: m = 2/3
Linear Functions Teacher m = 2/3 Make the rise and run bigger or  smaller to emphasize that there  are many places to find slope. y x 2 4 6 8 10 -2 -4 -6 -8 -10 2 Find 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.) run rise

Find the slope of the line going through the following points.

Find the slope of a line going through the following points:
Linear Functions Teacher m = -8/7 3 Find the slope of a line going through the following points: (-3, 5) and (4, -3)

Find the slope of the line going through the following points.
Teacher m = 3/5 Linear Functions 4 Find the slope of the line going through the following points. (0, 7) and (-5, 4)

Roofs Distance Height And many more...
Teacher Get ideas from students  where slope is used in the  "real world." Linear Functions Slope formula can be used to find the constant of  change in "real world" problems. Mountain Highways Roofs Growth Distance Height And many more...

Linear Functions When 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)

Teacher m = 50mph Linear Functions 5 If 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?

Teacher m = 7.5 meters per second Linear Functions 6 How many meters per second is a person running if they are at 10 meters in 3 seconds and 100 meters in 15 seconds?

Standard Form Point-Slope Form y = mx + b y - y1 = m(x - x1)
Linear Functions We are going to look at three different forms of the equations of lines. Each has its advantages and disadvantages in their uses. Slope-Intercept Form Standard Form Point-Slope Form y = mx + b Ax + By = C y - 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.

Determine the equation of the line from the given graph.
1) Determine the y-intercept. Remember, graphically, the y-intercept is where the graph crosses the y-axis. click b = -3 2) Graphically find the slope from any two points. click 3) Write the equation of the line using the slope-intercept form. y = mx + b y = 2x - 3 click

Which equation does this line represent?
Teacher D y = x - 6 Ask students which form it is in: Slope-intercept Linear Functions 7 Which equation does  this line represent? A y = -6x - 6 B y = -x - 6 C y = 6x + 6 D y = x - 6

Which graph represents the equation y = 3x - 2?
Linear Functions Teacher D line D A B C D 8 Which graph  represents the  equation y = 3x - 2? A Line A B Line B C Line C D Line D

What equation does line A represent?
9 What equation does line A represent? Teacher B A B C D A y = 2x + 3 B y = -2x + 3 C y = 0.5x + 3 D y = -0.5x + 3

What equation does line B represent?
10 What equation does line B represent? Teacher D A y = 2x + 3 A B C D B y = -2x + 3 C y = 0.5x + 3 D y = -0.5x + 3

Consider the equation 4x - 3y = 6.
Linear Functions Consider the equation 4x - 3y = 6. Which form is it in? Graph it using the  advantages of the form. Teacher It is in Standard Form. Graph  it using the x and y  intercepts. x-int: (3/2, 0) y-int: (0, -2)

Graph the equation 5x + 6y = -30 using the most appropriate method based on the form.
Teacher It is in Standard Form. Graph  it using the x and y  intercepts. x-int: (-6, 0) y-int: (0, -5)

Graph the equation 3x - 5y = -10 using the most appropriate method based on the form.
Teacher It is in Standard Form. Graph  it using the x and y  intercepts. x-int: (-10/3, 0) y-int: (0, 2)

Write 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.

Multiply each term by LCD: 15
Linear Functions 11 What is the Standard Form of: A 3x + 5y = 15 B 9x + 15y = 35 Teacher B: 9x + 15y = 35 Multiply each term by LCD: 15 C 15x - 9y = 35 D 5x - 3y = 15

Which form is this equation in? y - 3 = 4(x + 2)
12 Which form is this equation in? y - 3 = 4(x + 2) Point-Slope Form Teacher A Standard Form B Slope-Intercept Form C Point-Slope Form

13 Find y when x = 0. y - 3 = 4(x + 2) Point-Slope Form y is 11
Linear Functions 13 Find y when x = 0. y - 3 = 4(x + 2) Teacher Point-Slope Form y is 11

Put the following information for a line in Point-Slope form:
Teacher C y - 5 = -3(x + 2) Linear Functions 14 Put the following information for a line in Point-Slope form: m = -3 going through (-2, 5). A y - 3 = -2(x + 5) B y - 2 = 5(x - 3) C y - 5 = -3(x + 2) D y + 2 = -3(x - 5)

Put the following information for a line in Point-Slope form:
Linear Functions 15 Put the following information for a line in Point-Slope form: m = 2 going through (1, 4). Teacher C y - 4 = 2(x - 1) A y + 4 = 2(x + 1) B y + 4 = 2(x - 1) C y - 4 = 2(x - 1) D y - 4 = 2(x + 1)

Put the following information for a line in Point-Slope form:
Linear Functions 16 Put the following information for a line in Point-Slope form: m = -2/3 going through (-4, 3). Teacher D y - 3 = -2/3(x + 4) A y + 3 = -2/3(x + 4) B y - 3 = -2/3(x - 4) C y + 3 = -2/3(x - 4) D y - 3 = -2/3(x + 4)

Horizontal and Vertical Lines
Linear Functions Horizontal and Vertical Lines Example of Horizontal line y=2 Example of a Vertical Line x=3 Horizontal 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.

Which equation does the line represent?
Linear Functions Teacher C y = 4 Easy 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. 17 Which equation does  the line represent? A y = 4x B y = x + 4 C y = 4 D y = x

Teacher B Horizontal Linear Functions 18 Describe the slope of the following line: y = 4 (Think about which axis it "cuts.") A Vertical B Horizontal C Neither D Cannot be determined

Describe the slope of the following line: 2x - 3 = 4.
Linear Functions Teacher A  Vertical 19 Describe the slope of the following line: 2x - 3 = 4. A Vertical B Horizontal C Neither D Cannot be determined

Parallel and Perpendicular Lines
Parallel lines have the same slope: Perpendicular lines have opposite , reciprocal  slopes. Perpendicular lines have opposite , reciprocal  slopes. h(x) = x + 6 q(x) = x + 2 r(x) = x - 1 s(x) = x - 5 h(x) = -3x - 11 g(x) = 1/3x - 2

Drag the equation to complete the statement.
Linear Functions Drag the equation to  complete the statement. Teacher Have students come to the board and drag. A) y = -1/4x - 3 B) 1/5y = x - 2 C) y = 4x - 1 D) y = -1/5x + 9 E) 6x + y = 10 F) y = 1/5x G) y = 1/6x - 6 A) 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 _____________.

How 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. Distribute Rearrange the equation so that the y term is isolated on the right hand side. Make sure that y is always positive.

Are 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.

Both slopes are 2/5. Yes the lines are parallel.
Are the lines 2x + 5y = 12 and 8x + 20y = 16 parallel? Teacher Both slopes are 2/5. Yes the lines are parallel. Turn the equations into slope-intercept form.

Which line is perpendicular to y = -3x + 2?
Linear Functions Teacher B 20 Which line is perpendicular to y = -3x + 2? A y = -3x + 1 B C y = 5 D x = 2

Which line is perpendicular to y = 0?
Linear Functions Teacher D x = 2 21 Which line is perpendicular to y = 0? A y = -3x + 1 B y = x C y = 5 D x = 2

Teacher Slope-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 Functions Writing Equations of Lines Remember 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 Form Standard Form Point-Slope Form y = mx + b Ax + By = C y - y1 = m(x - x1)

Writing an equation in Slope- Intercept Form:
Linear Functions Writing 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

Write an equation in slope-intercept form through the given points.
Find the slope. Plug the coordinates from one point into x and y respectfully.

Linear Functions Teacher Write the equation of the line with slope of 1/2 and through  the point (2,5). Leave your answer in slope-intercept form.

Linear Functions Teacher Write the equation of the line through (-3,-2) and  perpendicular to y = -4/5x + 1 in slope-intercept form.

Linear Functions 22 Write the equation of the line going through the points (1, 2) and (3, -4). Your answer should be in slope-intercept form. Teacher A y = -3x + 5 y = -3x - 3 y = -3x + 2 B C D

Linear Functions Teacher 23 What is the equation of the line with a slope parallel to 3x + y = 2 going through the point (4, -2) A y = -3x + 10 B y = -3x - 14 C y = 3x + 10 D y = -3x + 14

Writing an equation in Point-Slope Form:
Linear Functions Writing 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).

Write an equation in point-slope form through the given two points.
Find the slope. Use either point.

Write the equation of the line through
Linear Functions Write the equation of the line through (5,6) and (7,1) using Point-Slope form. Teacher or

Linear Functions Write the equation of the line with slope of 1/2 and through the point (2,5) in Point-Slope form. Teacher

Linear Functions Teacher 24 Which of the following is the Point-Slope form of the line going through (-2, 4) and (6, -2)? A B C D

25 A y - 3 = 2(x - 3) B y = 2(x - 3) C y - 3 = 2x D y = 3
Teacher Linear Functions 25 Which of the following is the Point-Slope form of the  line going through (0, 3) having the slope of 2? A y - 3 = 2(x - 3) B y = 2(x - 3) C y - 3 = 2x D y = 3

Teacher 4x - y = 11 Have a class discussion about  the usefulness of this form and  how easy this was to do. Linear Functions Standard Form Write 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. Distribute Distribute Rearrange 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.

How to turn an equation from point-slope to standard form.
Teacher x + y = -6 Have 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.

Write the equation of the line through (-7, -2) and (1, 6) in Standard Form.
Teacher x - y = 5 Have a class discussion about  the usefulness of this form and  how easy this was to do.

Write the equation of the line with
Linear Functions Write the equation of the line with x-intercept of 5 and y-intercept of 10. Which  form would be easiest to use? Teacher y = -2x + 10 or y - 10 = -2x y = -2(x - 5) There is nothing  that states what  form you must  leave answer in,  so many answers  are correct.

Write the equation of the line through (4,1) and
Linear Functions Write the equation of the line through (4,1) and parallel to the line y = 3x Which form is easiest to  use in this case? Teacher y - 1 = 3(x - 4) y = 3x -11 3x - y = 11

Teacher This is an opinion question that  fuels discussion about different  forms and their purpose. Linear Functions 26 Given a point of (-3, 3) and a slope of -2, what is  the easiest form to write the equation in? A Slope-intercept Form B Point-Slope Form C Standard Form

Linear Functions 27 Given two points on a line, (3, 0) and (0, 5), what is the easiest form to write the equation in? Teacher This is an opinion question that  fuels discussion about different  forms and their purpose. A Slope-intercept Form B Point-Slope From C Standard Form

Linear Functions Teacher All of the answers are equations of  the line, but only A and B are in  Point-Slope form. 28 Which of the following is/are the equation(s) of the line through (1, 3) and (2, 5) in Point-Slope form? A y - 5 = 2(x - 2) B y - 3 = 2(x - 1) C y = 2x + 1 D 2x - y = -1

Exponential Functions

Exponential Functions
Goals and Objectives Students will be able to recognize and graph  exponential functions.

Exponential 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?

Exponential 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. X Y 1 10 2 20 3 30 4 40 Can we easily  predict what you would get if you  have 7 A's? \$\$ Number of A's

Exponential Functions
Teacher Capitalize 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 = 3x Graph it!! What would you get for 7 A's?

y = 3x Can you find the amount for 7 A's?
Exponential Functions Here is the graph of your function!  This is an example of exponential  growth. y = 3x Can you find the amount for 7 A's? Would that provide some motivation?

X Y 80 1 40 2 20 3 10 4 5 Now, let's look an opposite problem:
Teacher Draw the curve on the graph. Ask  students if the curve will ever go  below the x-axis and why? Exponential Functions Now, 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. X Y 80 1 40 2 20 3 10 4 5

Here is a graph of the M&M problem.
Teacher 1. 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 Functions Number of M&M's Days Here 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?

Exponential Functions
Now we are going to identify an  Exponential Function both  graphically and algebraically.

Stress and explore domain and range. Graphically
Teacher Stress and explore domain and range. Exponential Functions Graphically The 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 Growth Exponential Decay Why does the domain run from negative infinity to positive infinity? Why does the range run from 0 to positive infinity?

Which of the following are graphs of exponential growth?
Exponential Functions 29 Which of the following are graphs of exponential growth? (You can choose more than one.) Teacher A and F A B C D E F G H

Teacher D Exponential Functions 30 Which of the following are graphs of exponential decay? (You can choose more than one.) A B C D E F G H

The general form of an exponential function is
Teacher Students 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 Functions The general form of an exponential function is where x is the variable and a, b, and c are constants. b is the growth rate. If b > 1 then it is exponential growth If 0 < b < 1 then it is exponential decay y = c is the horizontal asymptote (0, a + c) is the y-intercept remember, to find the y-intercept, set x = 0.

Does the following exponential equation represent growth or decay?
Exponential Functions Teacher A Growth 31 Does the following exponential equation represent growth or decay? A Growth B Decay

Teacher C y = 4 Exponential Functions 32 For the same function, what is the equation of the horizontal  asymptote? A y = 2 B y = 3 C y = 4 D y = 5

Now, find the y-intercept:
Teacher C (0, 7) Exponential Functions 33 Now, find the y-intercept: A (0, 3) B (0, 4) C (0, 7) D (0, 9)

Exponential Functions
34 Now consider this new exponential equation:  Does it represent growth or decay? A Growth  B Decay Teacher B Decay

Which of the following is the equation of the horizontal asymptote?
Teacher C y = 3 Exponential Functions 35 Which of the following is the equation of the horizontal  asymptote? A y = 0.2 B y = 1 C y = 3 D y = 4

Find the y-intercept in the same function:
Exponential Functions 36 Find the y-intercept in the same function: A (0, 0.2) Teacher D (0, 4) B (0, 1) C (0, 3) D (0, 4)

B Decay. Rewrite the function with a positive exponent:
Teacher B Decay. Rewrite the function with  a positive exponent: Have a discussion on what is different  about this function. Exponential Functions 37 Does the following exponential function represent growth or decay? A Growth B Decay

Find the horizontal asymptote for the following function:
Exponential Functions Teacher A y = 0 38 Find the horizontal asymptote for the following function: A y = 0 B y = 1 C y = 3 D y = 4

For the same function, what is the y-intercept?
Exponential Functions Teacher D (0, 4) 39 For the same function, what is the y-intercept? A (0, 0) B (0, 1) C (0, 3) D (0, 4)

1) Identify horizontal asymptote (y = c)
Teacher Exponential Functions To 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 growth 3) Graph y-intercept (0,a+c) 4) Sketch graph Try it! Note: A horizontal asymptote is the horizontal line at y = c that the exponential function cannot pass.

Exponential Functions
Teacher Graph by hand:

Exponential Functions
Teacher Graph by hand:

Which of the following is a graph of:
Exponential Functions Teacher C 40 Which of the following is a graph of: A B D C

Which of the following is a graph of:
Teacher B Exponential Functions 41 Which of the following is a graph of: A B C D

Logarithmic Functions

Logarithmic Functions
Goals and Objectives Students 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.

Logarithmic 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.

A logarithmic function is the inverse of an exponential function.
Logarithmic Functions A 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.

Logarithmic 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.

Logarithmic Functions Logarithms 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 = answer logbaseanswer = exponent *A log that appears to have no base is automatically considered base 10. For example:

Logarithmic Functions baseexponent = answer logbaseanswer = exponent Teacher Rewrite the following in logarithmic form:

Logarithmic Functions baseexponent = answer logbaseanswer = exponent Teacher Rewrite the following in exponential form.

Logarithmic Functions
Teacher A 42 Which of the following is the correct logarithmic form of: A C B D

Logarithmic Functions
Teacher D 43 Which of the following is the correct logarithmic form of: A C B D

44 Which of the following is the correct exponential form of: B A B C
Teacher B Logarithmic Functions 44 Which of the following is the correct  exponential form of: A B C D

Which of the following is the correct exponential form of:
Logarithmic Functions Teacher C 45 Which of the following is the correct  exponential form of: A C B D

What is the exponential form of:
Logarithmic Functions Teacher A 46 What is the exponential form of: A B C D

Logarithmic 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

Logarithmic Functions
Teacher 47 Find c:

Teacher Logarithmic Functions 48 Find x:

Teacher Logarithmic Functions 49 Evaluate:

Logarithmic Functions
Teacher 4 50 Evaluate:

Teacher 3 Logarithmic Functions 51 Evaluate:

Use this as a reminder of negative exponents and what they mean. 52
Logarithmic Functions Teacher -4 Use this as a reminder of negative  exponents and what they mean. 52 Evaluate:

Use this as a reminder of negative exponents and what they mean.
Teacher -3 Use this as a reminder of negative  exponents and what they mean. Logarithmic Functions 53 Evaluate:

Teacher No solution. You cannot  take the log of a negative  number. Logarithmic Functions 54 Evaluate:

Teacher You need a calculator to do this  one. We will cover it in the next  few slides. Logarithmic Functions 55 Evaluate:

Teacher Logarithmic Functions Take out a calculator and find "log". What base is it? Use the calculator to find the following values. Round off to three decimal places.

Teacher 5.129 Make sure that students  close parentheses and  put them in the right  place. Logarithmic Functions Evaluate: 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.

Logarithmic Functions
Teacher 1.827 Make sure that students  close parentheses and  put them in the right  place. 56 Evaluate:

Logarithmic Functions
Teacher 1.011 Make sure that students  close parentheses and  put them in the right  place. 57 Evaluate:

58 Evaluate: -1.679 Logarithmic Functions
Teacher -1.679 Students 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. 58 Evaluate:

59 Evaluate: -0.900 Logarithmic Functions
Teacher Students 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. -0.900 Logarithmic Functions 59 Evaluate:

Algebraically, how would you solve:
Logarithmic Functions The change of base formula and converting to logarithms  will now allow you to solve more complicated exponential  equations. Teacher This 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:

Solve the following equation:
Logarithmic Functions Teacher Solve the following equation:

Solve the following equation.
Logarithmic Functions Teacher 60 Solve the following equation.

Logarithmic Functions
Teacher 61 Solve:

Logarithmic Functions
Teacher 62 Solve:

Logarithmic Functions
Practice using the same conversion to solve logarithmic  equations again. Teacher Solve:

Teacher Logarithmic Functions 63 Solve:

Teacher Logarithmic Functions 64 Solve:

Logarithmic Functions
Teacher 65 Solve:

Properties of Logarithms

Basic Properties of Logs
Properties of Logarithms Basic Properties of Logs

Properties of Logarithms
WHAT?

Relate logs to exponents...
Properties of Logarithms Relate logs to exponents... Take: In exponential form, it means: The property says that: And...

Relate logs to exponents...
Properties of Logarithms Relate logs to exponents... Take: In exponential form, it means: The property says that: And...

Relate logs to exponents...
Properties of Logarithms Relate logs to exponents... Take: In exponential form, it means: The property says that: And...

Relate logs to exponents...
Properties of Logarithms Relate logs to exponents... Take: In exponential form, it means: The property says that: And...

Relate logs to exponents...
Properties of Logarithms Relate logs to exponents... Take: In exponential form, it means: The property says that: And...

Examples: Use the Properties of Logs to expand:
Properties of Logarithms Teacher Examples: Use the Properties of Logs to expand:

Which choice is the expanded form of the following
Properties of Logarithms Teacher D 66 Which choice is the expanded form of the following A B C D

Which choice is the expanded form of the following
Properties of Logarithms 67 Which choice is the expanded form of the following Teacher C A B C D

Which choice is the expanded form of the following
Properties of Logarithms Teacher A 68 Which choice is the expanded form of the following A B C D

Which choice is the expanded form of the following
Properties of Logarithms 69 Which choice is the expanded form of the following Teacher C A B C D

Examples: Use the Properties of Logs to rewrite as a single log.
Properties of Logarithms Teacher Examples: Use the Properties of Logs to rewrite as a single log.

Contract the following logarithm:
Properties of Logarithms Teacher A 70 Contract the following logarithm: A B C D

Contract the following logarithmic expression:
Properties of Logarithms Teacher C 71 Contract the following logarithmic expression: A B C D

Which choice is the contracted form of the following:
Properties of Logarithms Teacher B 72 Which choice is the contracted form of the following: A B C D

Which choice is the contracted form of the following:
Teacher A Properties of Logarithms 73 Which choice is the contracted form of the following: A B C D

Solving Logarithmic Equations

Solving 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.

Solving Logarithmic Equations
Teacher This 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?

Solve: Solving Logarithmic Equations Teacher
The solution r = -1 is  actually an extraneous  solution as you cannot  take a log of x ≤ 0. Solving Logarithmic Equations Solve:

Solving Logarithmic Equations
Teacher Stress 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.

Solve the following equation:
Solving Logarithmic Equations Teacher 74 Solve the following equation:

Solve the following equation:
Teacher Solving Logarithmic Equations 75 Solve the following equation:

Solve the following equation:
Teacher Solving Logarithmic Equations 76 Solve the following equation:

Solve the following equation:
Solving Logarithmic Equations 77 Solve the following equation: Teacher ,

Solve the following equation:
Solving Logarithmic Equations Teacher 78 Solve the following equation:

Solve the following equation:
Solving Logarithmic Equations Teacher 79 Solve the following equation:

How can we use these concepts to solve:
Solving Logarithmic Equations Teacher It 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:

Try: Solving Logarithmic Equations
Teacher It is actually legal to  take the log of both  sides. You can then  apply the properties of  logarithms to solve. Try:

Solving Logarithmic Equations
Teacher 80 Solve:

Solving Logarithmic Equations
Teacher 81 Solve:

Solving Logarithmic Equations
Teacher 82 Solve:

Basic Properties of Natural Logs
Properties of Natural Logarithms Basic Properties of Natural Logs

Formally, e is defined to be: Simply, as a number:
e and ln The 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...

*Find ln on your calculator. This is loge.
e and ln As 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.

e and ln The graphs of e and ln are similar to our other functions.  The domain and range also remain the same.

e and ln e and natural logs have all of the same properties that other  exponentials and logarithms have. For example: because because

Write the following in the equivalent exponential or log form.
Teacher e and ln Write the following in the equivalent exponential or log form.

Put the following into a single logarithm:
e and ln Teacher Put the following into a single logarithm:

Expand the following logarithms:
Teacher e and ln Expand the following logarithms:

Put into a single logarithm:
Teacher D e and ln 83 Put into a single logarithm: A B C D

Expand the following logarithm:
Teacher B e and ln 84 Expand the following logarithm: A B C D

Expand the following logarithm:
e and ln Teacher B 85 Expand the following logarithm: A B C D

Put into a single logarithm:
Teacher A e and ln 86 Put into a single logarithm: A C B D

Solve the following equations:
e and ln Teacher *Sometimes you will be asked to leave answers as exact  numbers. Solve the following equations:

Solve the following equations:
Teacher *Only 60.6 or are solutions  because you  cannot take the  log of a  negative. e and ln Solve the following equations:

e and ln Teacher 87 Find the value of x.

e and ln Teacher 88 Find the value of x.

e and ln Teacher 89 Find the value of x.

e and ln Teacher 90 Find the value of x.

e and ln Teacher 91 Find the value of x.

e and ln 92 Find the value of x. Teacher

Growth and Decay Goals and Objectives Students will be able to model growth and decay  problems with exponential equations.

Growth and Decay Why 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.

Growth and Decay And...these functions will also model problems dealing  with something we ALL need to learn to work with... Money!

P = the principal (amount deposited)
Teacher \$ in 2 years. Growth and Decay This 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 years If \$500 is invested at 4% for 2 years, what will  account balance be?

It will take about 17.3 years for the money to double.
Growth and Decay Teacher It 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?

Growth and Decay 93 If \$1000 is invested at 4% for 3 years,  compounded continually, what is the account  balance? Teacher

Growth and Decay Teacher 19.8 years 94 If \$1000 is invested at 3.5%, how long  until the account balance is doubled?

Account Balance with Compound Interest:
Growth and Decay Teacher Spend time discussing parts  of each type of interest. There are other types of interest that you may encounter... I = interest P = principal (deposit) r = interest rate (decimal) t = time in years n = number of times compounded per unit of t A=account balance after the interest is included Simple Interest: Account Balance with Simple Interest SiSsSsss Account Balance with Compound Interest:

Teacher I = \$480 Growth and Decay Calculate 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.

Ask students which is the better deal.
Growth and Decay Teacher I = \$1600 Ask 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.

Teacher Growth and Decay Using the formula for compound interest, calculate the  amount in your account if you invest \$1300, over 15  years, at 4.6% interest compounded monthly.

Growth and Decay How 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

Growth and Decay 95 How 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

Teacher Growth and Decay 96 What was the rate of your investment if you invested \$5000 over 10 years, compounded continuously and you made \$2200 in interest?

Teacher Growth and Decay 97 How much was originally invested if you have \$63, in an account generating 4% interest (compounded monthly) over 15 years?

r = rate or growth or decay(decimal) t = time
Teacher Discuss the different uses of  the same formulas. Growth and Decay The 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

Example: A bacteria constantly grows at a rate of
Growth and Decay Teacher It will take the colony 23  hours to increase its  population from 100 to  1000. Example: A bacteria constantly grows at a rate of 10% per hour, if initially there were 100 how long till there were 1000?

Teacher The original value of  the car was about  \$30,345. Growth and Decay A 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?

A certain radioactive material has a half-life of 20 years.
Growth and Decay Teacher In 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.

Teacher It would take ≈ 2.7 hours Growth and Decay 98 If 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?

Teacher Growth and Decay 99 If you need your money to double in 8 years, what must the interest rate be if is  compounded quarterly?

Growth and Decay Teacher 100 NASA 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?

Growth and Decay Teacher 101 If the half-life of an element is 50 years, at what rate  does it decay?