Exponential Modeling and Logarithms

Students revisit exponential functions, including geometric sequences and series, and learn to manipulate logarithmic expressions and equations to solve problems involving exponential modeling.

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Unit Summary

Students have previously seen exponential functions in Algebra I. This unit builds off of that knowledge, revisiting exponential functions and including geometric sequences and series and continuous compounding situations. In the second part of the unit, students learn that the logarithm is the inverse of the exponent and to manipulate logarithmic expressions and equations. Finally, students apply their knowledge of logarithms to solve problems involving exponential modeling.

This unit is an excellent opportunity for students to practice mathematical modeling using exponential functions as models for situations in the world. Students will also look for and make use of structure as they manipulate logarithms and connect their knowledge of exponents to logarithms.

While this unit culminates the study of exponents and logarithms in the Common Core State Standards, it leads to essential topics in calculus. Students preparing to take a calculus course should emphasize algebraic manipulation of exponential and logarithmic expressions to rewrite them in a variety of ways, including using properties of logarithms and analyzing functions to connect their graphs to equations and contextual situations.

Pacing: 18 instructional days (16 lessons, 1 flex day, 1 assessment day)

Assessment

This assessment accompanies Unit 5 and should be given on the suggested assessment day or after completing the unit.

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Unit Prep

Intellectual Prep

Internalization of Standards via the Unit Assessment

Take unit assessment. Annotate for:
- Standards that each question aligns to
- Purpose of each question: spiral, foundational, mastery, developing
- Strategies and representations used in daily lessons
- Relationship to Essential Questions of unit
- Lesson(s) that assessment points to

Internalization of Trajectory of Unit

Read and annotate "Unit Summary."
Notice the progression of concepts through unit using the “Unit at a Glance.”
Do all target tasks. Annotate the target tasks for:
- Essential questions
- Connection to assessment questions.
Answer the essential questions. (In the beginning, submit them to your instructional leader; toward the end, just bring them to the meeting).

Unit-Specific Intellectual Prep

Watch this video, “How I feel about Logarithms” by Vi Hart.

Essential Understandings

Write expressions or equations or change the form of an equation to model contexts and connect components of those expressions and equations to graphs and contexts.
Explain that logarithmic functions are the inverse of exponential functions and have parallel properties to exponents used to rewrite expressions and solve equations. Know that the properties of logarithms can be used to change the form of equations to reveal solution paths.
Use knowledge of exponents and logarithms to solve exponential modeling problems where a piece of information is missing.

Vocabulary

Geometric sequence	Geometric series
Compounding/Continuous compounding	Percentage Rate
e (Euler's number)	Base
Rate	Argument
Principal	Finite geometric series
Sum of geometric series	Summation notation $${\sum}$$
Logarithm	Natural log
Common log	Change of base
Product property of logarithms (Logarithm law)	Quotient property (Logarithm law)
Power property (Logarithm law)	Exponentiate
Exponential growth	Exponential decay

Materials

Equations, tables, graphs, and contextual situations
A calculator or other technology to graph and solve problems using exponential modeling and logarithms

Lesson Map

Topic A: Modeling with and Interpreting Exponential Functions

1

F.IF.A.3

F.IF.B.5

F.LE.A.2

Identify, model, and analyze geometric sequences.

2

F.IF.B.4

F.LE.A.2

F.IF.C.8.B

Analyze and construct exponential functions that model contexts.

3

F.LE.B.5

F.BF.A.1.A

Write and change the form of exponential functions that model compounding interest.

4

A.SSE.B.3.C

F.BF.A.1.A

Define and use $$e$$ in continuous compounding situations.

5

A.SSE.B.4

Describe the derivation of the formula for the sum of a finite geometric series and use it to solve problems.

6

A.SSE.B.4

Find the sum of an infinite geometric series.

Topic B: Definition and Meaning of Logarithms

7

F.LE.A.4

Describe and evaluate simple numeric logarithms (Part I).

8

F.LE.A.4

Describe and evaluate simple numeric logarithms (Part II).

9

F.BF.B.3

F.BF.B.5

F.IF.C.7.E

F.BF.B.4.B

F.BF.B.4.C

Describe logarithms as the inverse of exponential functions and graph logarithmic functions.

10

F.LE.A.4

F.BF.B.4.C

Evaluate common and natural logs using tables, graphs, and calculators.

11

F.LE.A.4

Understand and apply the change of base property to evaluate logarithms.

12

F.LE.A.4

Develop and use the product and quotient properties of logarithms to write equivalent expressions.

13

F.LE.A.4

F.BF.B.4.B

Develop and use the power property of logarithms to write equivalent expressions.

14

F.LE.A.4

Solve equations with logarithms.

15

F.LE.A.4

A.SSE.A.1.B

Use logarithms to solve exponential modeling problems (Part I).

16

F.LE.A.4

A.SSE.A.1.B

Use logarithms to solve exponential modeling problems (Part II).

Common Core Standards

Key: Major Cluster Supporting Cluster Additional Cluster

Core Standards

Building Functions

F.BF.A.1 — Write a function that describes a relationship between two quantities Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Building Functions

F.BF.A.1 — Write a function that describes a relationship between two quantities Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
F.BF.A.1.A — Determine an explicit expression, a recursive process, or steps for calculation from a context.

Building Functions

F.BF.A.1.A — Determine an explicit expression, a recursive process, or steps for calculation from a context.
F.BF.B.3 — Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), 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. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

Building Functions

F.BF.B.3 — Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), 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. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
F.BF.B.4 — Find inverse functions.

Building Functions

F.BF.B.4 — Find inverse functions.
F.BF.B.4.B — Verify by composition that one function is the inverse of another.

Building Functions

F.BF.B.4.B — Verify by composition that one function is the inverse of another.
F.BF.B.4.C — Read values of an inverse function from a graph or a table, given that the function has an inverse.

Building Functions

F.BF.B.4.C — Read values of an inverse function from a graph or a table, given that the function has an inverse.
F.BF.B.5 — Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

Building Functions

F.BF.B.5 — Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

Interpreting Functions

F.IF.A.3 — Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.

Interpreting Functions

F.IF.A.3 — Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.
F.IF.B.4 — For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Interpreting Functions

F.IF.B.4 — For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
F.IF.B.5 — Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Interpreting Functions

F.IF.B.5 — Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
F.IF.C.7 — Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Interpreting Functions

F.IF.C.7 — Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
F.IF.C.7.E — Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

Interpreting Functions

F.IF.C.7.E — Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
F.IF.C.8 — Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

Interpreting Functions

F.IF.C.8 — Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
F.IF.C.8.B — Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)^t, y = (0.97)^t, y = (1.01 ^12t, y = (1.2)^t/10, and classify them as representing exponential growth or decay.

Interpreting Functions

F.IF.C.8.B — Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)^t, y = (0.97)^t, y = (1.01 ^12t, y = (1.2)^t/10, and classify them as representing exponential growth or decay.

Linear, Quadratic, and Exponential Models

F.LE.A.2 — Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

Linear, Quadratic, and Exponential Models

F.LE.A.2 — Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
F.LE.A.4 — For exponential models, express as a logarithm the solution to ab^ct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

Linear, Quadratic, and Exponential Models

F.LE.A.4 — For exponential models, express as a logarithm the solution to ab^ct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.
F.LE.B.5 — Interpret the parameters in a linear or exponential function in terms of a context.

Linear, Quadratic, and Exponential Models

F.LE.B.5 — Interpret the parameters in a linear or exponential function in terms of a context.

Seeing Structure in Expressions

A.SSE.A.1 — Interpret expressions that represent a quantity in terms of its context Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Seeing Structure in Expressions

A.SSE.A.1 — Interpret expressions that represent a quantity in terms of its context Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
A.SSE.A.1.B — Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)ⁿ as the product of P and a factor not depending on P.

Seeing Structure in Expressions

A.SSE.A.1.B — Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)ⁿ as the product of P and a factor not depending on P.
A.SSE.B.3 — Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Seeing Structure in Expressions

A.SSE.B.3 — Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
A.SSE.B.3.C — Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15^t can be rewritten as (1.151/12)^12t 1.012^12t to reveal the approximate equivalent monhly interest rate if the annual rate is 15%.

Seeing Structure in Expressions

A.SSE.B.3.C — Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15^t can be rewritten as (1.151/12)^12t 1.012^12t to reveal the approximate equivalent monhly interest rate if the annual rate is 15%.
A.SSE.B.4 — Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Seeing Structure in Expressions

A.SSE.B.4 — Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Foundational Standards

High School — Number and Quantity

N.RN.A.1

High School — Number and Quantity

N.RN.A.1 — Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5^1/3 to be the cube root of 5 because we want (5^1/3)³ = 5(^1/3)³ to hold, so (5^1/3)³ must equal 5.
N.RN.A.2

High School — Number and Quantity

N.RN.A.2 — Rewrite expressions involving radicals and rational exponents using the properties of exponents.

Linear, Quadratic, and Exponential Models

F.LE.A.1

Linear, Quadratic, and Exponential Models

F.LE.A.1 — Distinguish between situations that can be modeled with linear functions and with exponential functions.
F.LE.A.1.A

Linear, Quadratic, and Exponential Models

F.LE.A.1.A — Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
F.LE.A.1.B

Linear, Quadratic, and Exponential Models

F.LE.A.1.B — Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
F.LE.A.1.C

Linear, Quadratic, and Exponential Models

F.LE.A.1.C — Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

Reasoning with Equations and Inequalities

A.REI.A.1

Reasoning with Equations and Inequalities

A.REI.A.1 — Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

Seeing Structure in Expressions

A.SSE.A.2

Seeing Structure in Expressions

A.SSE.A.2 — Use the structure of an expression to identify ways to rewrite it. For example, see x⁴ — y⁴ as (x²)² — (y²)², thus recognizing it as a difference of squares that can be factored as (x² — y²)(x² + y²).

Standards for Mathematical Practice

CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.
CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.
CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.
CCSS.MATH.PRACTICE.MP4 — Model with mathematics.
CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.
CCSS.MATH.PRACTICE.MP6 — Attend to precision.
CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.
CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.

Rational and Radical Functions

Unit Circle and Trigonometric Functions

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Exponential Modeling and Logarithms

Unit Summary

Assessment

Unit Prep

Intellectual Prep

Essential Understandings

Vocabulary

Materials

Lesson Map

Topic A: Modeling with and Interpreting Exponential Functions

1

2

3

4

5

6

Topic B: Definition and Meaning of Logarithms

7

8

9

10

11

12

13

14

15

16

Common Core Standards

Core Standards

Building Functions

Building Functions

Building Functions

Building Functions

Building Functions

Building Functions

Building Functions

Building Functions

Interpreting Functions

Interpreting Functions

Interpreting Functions

Interpreting Functions

Interpreting Functions

Interpreting Functions

Interpreting Functions

Interpreting Functions

Linear, Quadratic, and Exponential Models

Linear, Quadratic, and Exponential Models

Linear, Quadratic, and Exponential Models

Linear, Quadratic, and Exponential Models

Seeing Structure in Expressions

Seeing Structure in Expressions

Seeing Structure in Expressions

Seeing Structure in Expressions

Seeing Structure in Expressions

Seeing Structure in Expressions

Foundational Standards

High School — Number and Quantity

High School — Number and Quantity

High School — Number and Quantity

Linear, Quadratic, and Exponential Models

Linear, Quadratic, and Exponential Models

Linear, Quadratic, and Exponential Models

Linear, Quadratic, and Exponential Models

Linear, Quadratic, and Exponential Models

Reasoning with Equations and Inequalities

Reasoning with Equations and Inequalities

Seeing Structure in Expressions

Seeing Structure in Expressions

Standards for Mathematical Practice

Unit 4

Rational and Radical Functions

Unit 6

Unit Circle and Trigonometric Functions