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New Jersey Center for Teaching and Learning

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1 Click to go to website: www.njctl.org
New Jersey Center for Teaching and Learning Progressive Mathematics Initiative This material is made freely available at 
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used for any commercial purpose without the written 
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2 Linear, Exponential and Logarithmic Functions
Algebra II Linear, Exponential and Logarithmic Functions

3 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

4 Linear Functions Return to Table of Contents

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

6 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.

7 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

8 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 =

9 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 =

10 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)?

11 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

12 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 =

13 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

14 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

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

16 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)

17 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)

18 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...

19 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)

20 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?

21 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?

22 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.

23 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

24 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

25 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

26 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

27 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

28 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)

29 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)

30 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)

31 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.

32 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

33 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

34 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

35 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)

36 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)

37 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)

38 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.

39 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

40 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

41 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

42 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

43 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 _____________.

44 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.

45 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.

46 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.

47 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

48 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

49 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)

50 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

51 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.

52 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.

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

54 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

55 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

56 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).

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

58 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

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

60 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

61 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

62 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.

63 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.

64 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.

65 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.

66 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

67 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

68 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

69 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

70 Exponential Functions
Return to Table of Contents

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

72 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?

73 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

74 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?

75 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?

76 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

77 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?

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

79 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?

80 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

81 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

82 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.

83 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

84 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

85 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)

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

87 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

88 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)

89 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

90 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

91 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)

92 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.

93 Exponential Functions
Teacher Graph by hand:

94 Exponential Functions
Teacher Graph by hand:

95 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

96 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

97 Logarithmic Functions
Return to Table of Contents

98 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.

99 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.

100 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.

101 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.

102 baseexponent = answer logbaseanswer = exponent
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:

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

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

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

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

107 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

108 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

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

110 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

111 Logarithmic Functions
Teacher 47 Find c:

112 Teacher Logarithmic Functions 48 Find x:

113 Teacher Logarithmic Functions 49 Evaluate:

114 Logarithmic Functions
Teacher 4 50 Evaluate:

115 Teacher 3 Logarithmic Functions 51 Evaluate:

116 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:

117 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:

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

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

120 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.

121 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.

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

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

124 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:

125 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:

126 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:

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

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

129 Logarithmic Functions
Teacher 61 Solve:

130 Logarithmic Functions
Teacher 62 Solve:

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

132 Teacher Logarithmic Functions 63 Solve:

133 Teacher Logarithmic Functions 64 Solve:

134 Logarithmic Functions
Teacher 65 Solve:

135 Properties of Logarithms
Return to Table of Contents

136 Basic Properties of Logs
Properties of Logarithms Basic Properties of Logs

137 Properties of Logarithms
WHAT?

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

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

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

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

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

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

144 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

145 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

146 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

147 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

148 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.

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

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

151 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

152 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

153 Solving Logarithmic Equations
Return to Table of Contents

154 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.

155 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?

156 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:

157 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.

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

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

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

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

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

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

164 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:

165 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:

166 Solving Logarithmic Equations
Teacher 80 Solve:

167 Solving Logarithmic Equations
Teacher 81 Solve:

168 Solving Logarithmic Equations
Teacher 82 Solve:

169 e and ln Return to Table of Contents

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

171 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...

172 *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.

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

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

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

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

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

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

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

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

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

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

183 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:

184 e and ln Teacher 87 Find the value of x.

185 e and ln Teacher 88 Find the value of x.

186 e and ln Teacher 89 Find the value of x.

187 e and ln Teacher 90 Find the value of x.

188 e and ln Teacher 91 Find the value of x.

189 e and ln 92 Find the value of x. Teacher

190 Growth and Decay Return to Table of Contents

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

192 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.

193 Growth and Decay And...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 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?

195 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?

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

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

198 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:

199 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.

200 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.

201 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.

202 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

203 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

204 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?

205 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?

206 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

207 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?

208 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?

209 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.

210 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?

211 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?

212 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?

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


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