Course Name: Algebra 2/Math III
Unit # 5
Unit Title: Logarithmic and Exponential Functions
Enduring understanding (Big Idea): Students will understand how to identify and graph exponential and logarithmic functions,
recognize that they are inverses of one another, and be aware of the types of changes in the graphs brought on by change in base and
by other transformations. Students will also understand how to use logarithmic and exponential functions to model real world
exponential growth and decay situations.
Essential Questions: How do you model a quantity that changes regularly over time by the same percentage? What kinds of things in
life can exponential/logarithmic equations help us do? Explain why an exponential model eventually exceeds a quantity increasing
linearly, quadratically, or (more generally) as a polynomial function.
BY THE END OF THIS UNIT:
Students will know…
Students will be able to:
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That a logarithmic function is the inverse of an
exponential function.
The relationships between logarithmic and exponential
functions.
The differences between the functions for exponential
growth and decay observed algebraically (i.e. y=abx, a is
the y-intercept, 0<b<1 represents decay, b>1 represents
growth). Be able to explain why these are the properties
by using the table of values for the corresponding
exponential function.
When given the graph of an exponential function, if the
function represents exponential growth or decay.
The horizontal asymptotes, x and y intercepts, and end
behavior of exponential and logarithmic functions.
The change of base formula.
That a logarithm with base 10 is the common logarithm
and a logarithm with base e is the natural logarithm.
That the natural logarithm is expressed as lnx and has all
of the same properties as logbx.
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Vocabulary:
Exponential function, exponential growth, exponential decay,
growth factor, decay factor, base e, continuously compounded
interest, logarithm, logarithmic function, inverse, compression,
stretch, horizontal and vertical shift, parent function, product
property, quotient property, power property, change of base
formula, exponential equation, common logarithm, and natural
logarithmic function.
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
PAGE 1
Graph exponential and logarithmic functions without
technology by shifting the parent functions.
Explain in words relationships between various
transformations and base changes in the equations.
Use properties of logarithmic functions to solve
logarithmic and exponential equations.
Use technology to solve logarithmic and exponential
equations.
Explain why a real-world situation is represented
exponentially as opposed to linearly or quadratically.
How to express a logarithmic function as an exponential
function and vice versa.
How to graph exponential and logarithmic functions with
technology.
How to shift exponential and logarithmic parent functions
to form new functions.
How to expand logarithms algebraically and graphically.
How to manipulate and apply logarithmic functions.
Determine the growth or decay factor of an exponential
function or situation.
Write an exponential function given a growth or decay
situation using y=a(1+r)t.
Write an exponential function for continuously
compounded interest using y-aert.
Course Name: Algebra 2/Math III
Unit # 5
Unit Resources:
Unit Title: Logarithmic and Exponential Functions
Mathematical Practices in Focus:
1. Make sense of problems and persevere in solving.
Algebra 2 textbook (Pearson), accompanying website:
www.pearsonsuccessnet.com, and NCDPI common core
unpacking documents.
2. Reason abstractly.
3. Construct viable arguments.
4. Model with Mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
CCSS-M Included:
F.BF.1, F.BF. 3, F.BF. 4, F.IF.7, F.IF.9, F.LE.3, F.LE.4, A.REI.10
Suggested Pacing:
10 days
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
PAGE 2
Course Name: Algebra 2/Math III
Unit # 5
Unit Title: Logarithmic and Exponential Functions
CORE CONTENT
Cluster Title: Build a function that models a relationship between two quantities and construct and compare linear, quadratic, and
exponential models.
Standards:
F.BF.1 Write a function that describes a relationship between two quantities.
a. Determine an explicit expression, a recursive process, or steps for calculation from a context.
b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a
cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.
Concepts and Skills to Master:
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Determine the growth or decay factor of an exponential function or situation.
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Write an exponential function given a growth or decay situation using y=a(1+r)t.
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Write an exponential function for continuously compounded interest using y-aert.
SUPPORTS FOR TEACHERS
Critical Background Knowledge:
 Standard form of exponential functions (y=abx) and the meanings of a and b.
 Base e
 How to interpret an exponential growth or decay situation and express it using y=a(1+r)t.
 How to interpret continuously compounded interest in a situation and express it using y=aert.
 How to convert percentages to decimals.
Academic Vocabulary:
Exponential function, exponential growth, exponential decay, growth factor, decay factor, base e, continuously compounded interest.
Suggested Instructional Strategies:
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Have students develop the formula y=a(1+r)t by
calculating a table of values. Be sure that they can
interpret the situation as growth or decay.
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Be sure to give students a variety of examples of
different growth and decay situations.
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Be sure to give students a variety of examples and
problems where they are asked to solve for different
amounts, time, rates, etc.
Resources:
Algebra 2 Textbook Correlation:
7-1: example problems 2 and 3
Pg. 439 #26-44
7-2: example problem 5
Pg. 448 #28-32, 36-37
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
PAGE 3
Course Name: Algebra 2/Math III
Unit # 5
Sample Assessment Tasks
Skill-based task:
Unit Title: Logarithmic and Exponential Functions
Problem Task:
1. Find the amount in a continuously compounded account
after 3 years if the principal is $2000 and the annual
interest rate is 5.1%.
1. Performance Task 1 from Chapter 7 Pearson teaching
resources (www.pearsonsuccessnet.com).
2. See textbook problems from above textbook correlation.
CORE CONTENT
Cluster Title: Construct and compare linear, quadratic and exponential models and solve problems.
Standards:
F.LE.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly,
quadratically, or (more generally) as a polynomial function.
Concepts and Skills to Master:
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Given a table of data and/or a situation, understand why an exponential function is used to model the data and/or situation as
opposed to a linear or quadratic.
SUPPORTS FOR TEACHERS
Critical Background Knowledge:
 Domain and range.
 Standard form of exponential functions (y=abx) and the meanings of a and b.
 Graphing a function from a table.
 Properties of exponential, linear, quadratic, and other polynomial functions (number of turns, increasing/decreasing, etc.).
Academic Vocabulary:
Exponential function, exponential growth, exponential decay, growth factor, decay factor
Suggested Instructional Strategies:
•
Have students create exponential functions by first giving
them a table of values. Have the students plot the points
and then determine the growth or decay factor. Have
students explain why the table of values should not be
represented by a linear , quadratic, or other polynomial
function. BE SURE that they understand the differences
and can explain WHY exponential data is different from
linear and quadratic.
Resources:
Algebra 2 Textbook Correlation:
7-1: example problems 1 and 4
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
PAGE 4
Course Name: Algebra 2/Math III
•
Unit # 5
Unit Title: Logarithmic and Exponential Functions
Have students come up with their own situation and data
that represents exponential, linear, quadratic, and other
polynomial functions. Have them explain the differences
and the reasons for their choices.
Sample Assessment Tasks
Skill-based task:
Problem Task:
1. Performance Task 1 from Chapter 7 Pearson teaching
resources (www.pearsonsuccessnet.com). Have
students explain why the graph is exponential and not
linear or quadratic.
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
PAGE 5
Course Name: Algebra 2/Math III
Unit # 5
Unit Title: Logarithmic and Exponential Functions
CORE CONTENT
Cluster Title: Analyze functions using different representations.
Standards:
F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by
verbal description).
Concepts and Skills to Master:
•
Graph exponential and logarithmic functions by plotting points from a table.
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The differences between the functions for exponential growth and decay observed algebraically (i.e. y=abx, a is the yintercept, 0<b<1 represents decay, b>1 represents growth). Be able to explain why these are the properties by using the
table of values for the corresponding exponential function.
•
Given the graph of an exponential function, determine if the function represents exponential growth or decay.
SUPPORTS FOR TEACHERS
Critical Background Knowledge:
 Domain and range
 Standard form of exponential functions (y=abx) and the meanings of a and b.
 How to interpret an exponential growth or decay situation and express it using y=a(1+r)t.
Academic Vocabulary:
Exponential function, exponential growth, exponential decay, growth factor, decay factor, asymptotes
Suggested Instructional Strategies:
•
Have students compare the equations, tables, and
graphs of linear, quadratic, and other polynomials to
those of exponential functions.
Sample Assessment Tasks
Skill-based task:
Resources:
Algebra 2 Textbook Correlation:
7-1: example problem 1 and the notes on page 435.
Pg. 439 #1-25, 45
Problem Task:
1. Does y=2(1.3)x represent growth or decay? What is the
y-intercept? What is the growth or decay factor?
2. Given a graph of an exponential function, determine if it
is growth or decay.
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
PAGE 6
1. Performance Task 1 from Chapter 7 Pearson teaching
resources (www.pearsonsuccessnet.com).
2. See problems from above textbook correlation.
Course Name: Algebra 2/Math III
Unit # 5
Unit Title: Logarithmic and Exponential Functions
CORE CONTENT
Cluster Title: Build new functions from existing functions.
Standard:
F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) = + k, kf(x), and f(x+k) for specific values of k (both positive and
negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using
technology.
Concepts and Skills to Master:
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Graph y=bx and observe it as the parent exponential function, then graph y=abx and observe how the value of a either
stretches or compresses the graph of y=bx.
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Graph y=abx and y=ab(x-h) and observe that y=ab(x-h) is the same as the vertical stretch or compression of y=(ab-h)bx.
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Observe that y=abx +k shifts the horizontal asymptote from y=0 to y=k.
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Graph y=logbx as the parent logarithmic function, then graph y=alogb(x-h) + k and observe: 1) how the value of a either
stretches or compresses the graph of y=logbx and 2) the vertical shift of y=logbx by h and the horizontal shift of y=logbx by k.
SUPPORTS FOR TEACHERS
Critical Background Knowledge:
Parent graphs of exponential and logarithmic functions as f(x) and how f(x) + k, af(x), and f(x=k) affects the parent function.
Academic Vocabulary:
Exponential function, logarithmic function, inverse, compression, stretch, horizontal and vertical shift, parent function
Suggested Instructional Strategies:

Technology should be used when first learning the above
skills and concepts. HOWEVER, once the students have
mastered the above skills and concepts, they should be
able to graph any exponential and logarithmic function
without the use of technology. Assessment of mastery
should be done without the use of technology (i.e. graph
without a calculator).
Resources:
Algebra 2 Textbook Correlation:
7-2 pgs. 442-444 example problems 1-2
Pg. 447 #1-5, 7-21
7-3 pg. 445 example problem 5
Pg. 456 #40-43, 69-76
 This standard should be taught in two sections:
1. The exponential functions should be taught with, or after,
the standard F.BF.1. Students should understand the
parent function first (F.BF.1) then learn how to stretch,
compress, and shift.
2. The logarithmic parent function, stretching, compressing,
and shifting of the parent function should be taught after
the logarithmic function is introduced as the inverse of
the exponential function (F.LE.3 and F.BF.4).
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
PAGE 7
Course Name: Algebra 2/Math III
Sample Assessment Tasks
Skill-based task:
1. Graph y=2x + 5 by hand.
2. Graph y=log4(x-1) + 3 by hand.
Unit # 5
Unit Title: Logarithmic and Exponential Functions
Problem Task:
7-2 page 449 #38-41
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
PAGE 8
Course Name: Algebra 2/Math III
Unit # 5
Unit Title: Logarithmic and Exponential Functions
CORE CONTENT
Cluster Title: Analyze functions using different representations.
Standard:
F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for
more complicated cases.
e. Graph exponential and logarithmic functions, showing intercepts and end behavior.
Concepts and Skills to Master:
•
Graph y=bx and observe it as the parent exponential function, then graph y=abx and observe how the value of a either
stretches or compresses the graph of y=bx.
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Observe that a in the above graph is the y-intercept. Interpret this as the starting value in context.
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Observe that y=abx +k shifts the horizontal asymptote from y=0 to y=k. Observe how this shift affects the y-intercept.
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Graph y=logbx as the parent logarithmic function, then graph y=alogb(x-h) + k and observe: 1) how the value of a either
stretches or compresses the graph of y=logbx and 2) the vertical shift of y=logbx by h and the horizontal shift of y=logbx by k.
SUPPORTS FOR TEACHERS
Critical Background Knowledge:
Parent graphs of exponential and logarithmic functions as f(x) and how f(x) + k, af(x), and f(x=k) affects the parent function.
Academic Vocabulary:
Exponential function, logarithmic function, compression, stretch, horizontal and vertical shift, parent function, end behavior, x and y
intercepts.
Suggested Instructional Strategies:

Technology should be used when first learning the above
skills and concepts. HOWEVER, once the students have
mastered the above skills and concepts, they should be
able to graph any exponential and logarithmic function
without the use of technology. Assessment of mastery
should be done without the use of technology (i.e. graph
without a calculator).

The students should be able to explain the meaning of
the constants and coefficients in the context of a word
problem.

This standard should be broken up into exponential
functions and logarithmic functions. The exponential
functions should be taught with the standard F.BF.3 and
Resources:
Algebra 2 Textbook Correlation:
7-1 pg. 435, example problem 2
Pg. 439 #18-25
7-2 pg. 442
Pg. 448 #35
7-3 pg. 454, example problem 4
Pg. 457 #69-72
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
PAGE 9
Course Name: Algebra 2/Math III
Unit # 5
Unit Title: Logarithmic and Exponential Functions
the logarithmic functions taught with standard F.BF.4.
Sample Assessment Tasks
Skill-based task:
Problem Task:
See above textbook correlations.
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
PAGE 10
Course Name: Algebra 2/Math III
Unit # 5
Unit Title: Logarithmic and Exponential Functions
CORE CONTENT
Cluster Title: Build new functions from existing functions.
Standard:
F.BF.4 Find inverse functions
Concepts and Skills to Master:
• Understand that the logarithmic function is the inverse of the exponential function.
• The exponential function y=bx is one-to-one and the inverse x=by is a function. Express x=by as y=logbx.
• Write an exponential equation as a logarithmic equation and a logarithmic equation as an exponential equation.
• Evaluate a logarithmic equation by writing it as an exponential equation.
SUPPORTS FOR TEACHERS
Critical Background Knowledge:
•
•
Logarithmic form and exponential form of an equation.
Graphically the logarithmic function is the inverse of the exponential function.
Academic Vocabulary:
Logarithm, logarithmic function, inverse, exponential function, domain and range.
Suggested Instructional Strategies:
1. INVERSE functions must be taught before this unit.
2. Introduce the logarithmic function as the inverse by
having students graph the inverse with a table first, then
convert the exponent form into the log form.
Sample Assessment Tasks
Skill-based task:
Resources:
Algebra 2 Textbook Correlation:
7-3 pg. 451, example problems 1 and 2, pg. 454, example
problem 4
Pg. 456 #12-31, 77-79
Problem Task:
1. Performance Task 4 from Chapter 7 Pearson teaching
resources (www.pearsonsuccessnet.com).
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
PAGE 11
Course Name: Algebra 2/Math III
Unit # 5
Unit Title: Logarithmic and Exponential Functions
CORE CONTENT
Cluster Title: Construct and compare linear, quadratic, and exponential models and solve problems.
Standard:
F.LE.4 For exponential models, express as a logarithm the solution to abct=d where a, c, and d are numbers and the base is 2, 10, or e;
evaluate the logarithm using technology.
Concepts and Skills to Master:
 Properties of logarithms (product, quotient, power).
 Expand and compress logarithms using the properties.
 Change of base formula.
 A logarithm with base 10 is the common logarithm and a logarithm with base e is the natural logarithm.
 The natural logarithm is expressed as lnx and has all of the same properties as logbx.
 Solve logarithmic equations using the properties.
SUPPORTS FOR TEACHERS
Critical Background Knowledge:

Properties of logarithms.

How to solve logarithmic equations.

Change of base formula.
Academic Vocabulary:
Logarithm, logarithmic function, inverse, product property, quotient property, power property, change of base formula, exponential
equation, common logarithm, and natural logarithmic function.
Suggested Instructional Strategies:
1. Be sure that the students understand that the logarithm
is a function and that logarithm equations must be solved
by using the properties. Many students will see log(x)=4
and try to solve by dividing by the word “log”. Be sure to
correct this misunderstanding and constantly reinforce
that log(x) is a function, not “log” times x.
Resources:
Algebra 2 Textbook Correlation:
7-3 pg. 453, example problem 3
7-4 pgs. 462-465, example problems 1-4
Pg. 466 #9-44, 50-72, 75-79
7-5 pgs. 469-472, example problems 1-6
Pg. 473 #7-45, 59-61, 64-75, 79-81
7-6 pgs. 478-479, example problems 1-3
Pg. 481 #11-37, 42-51, 54-56, 60-63
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
PAGE 12
Course Name: Algebra 2/Math III
Unit # 5
Sample Assessment Tasks
Skill-based task:
1.
2.
3.
4.
5.
6.
Unit Title: Logarithmic and Exponential Functions
Problem Task:
Express log25 + log26 as a single logarithm.
𝟐
Expand ln( ).
𝟑
Solve 5x=23
Solve log (4x-3)=2
Solve ln(2x)=5
Solve e3x=18
1. Performance Tasks 2 and 3 from Chapter 7 Pearson
teaching resources (www.pearsonsuccessnet.com).
2. See above textbook correlation.
CORE CONTENT
Cluster Title: Represent and solve equations and inequalities graphically.
Standard:
A.REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane.
Concepts and Skills to Master:

SUPPORTS FOR TEACHERS
Critical Background Knowledge:

Academic Vocabulary:
Suggested Instructional Strategies:
Resources:
Algebra 2 textbook correlation:
.
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
PAGE 13
Course Name: Algebra 2/Math III
Sample Assessment Tasks
Skill-based task:
Unit # 5
Unit Title: Logarithmic and Exponential Functions
Problem Task:
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
PAGE 14