Lesson 7 | Exponential Modeling and Logarithms | 11th Grade Mathematics

Objective

Describe and evaluate simple numeric logarithms (Part I).

Common Core Standards

Core Standards

The core standards covered in this lesson

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 abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

Foundational Standards

The foundational standards covered in this lesson

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 51/3 to be the cube root of 5 because we want (51/3)³ = 5(1/3)³ to hold, so (51/3)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.

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

Describe the pattern in converting between logarithms and exponential equations and describe the purpose of logs.
Write logs as exponents and exponents as logs.
Evaluate numeric logarithms to whole numbers and prove using exponents.
Identify and explain domain restrictions of logarithms.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

Watch the video “How I Feel About Logarithms” by Vi Hart.
Be sure to reference the Richter scale, pH, and decibels as real-life applications of the logarithmic scale.
Students may need to review solving equations of the form $${2^x=8}$$ before they can fully access this lesson.
This lesson focuses on whole numbers. Note that the second anchor problem has three examples of negative or fractional arguments; this can serve as a preview of the next lesson, but there is no need to spend too much time on these.

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Anchor Problems

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Problem 1

A colony of bacteria starts with 5 cells and grows so that it doubles in size every hour. How long will it be until there are 160 bacteria? How long will it be until there are 1,000 bacteria?

Guiding Questions

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Problem 2

Take a guess at what these statements are saying:

$${{{power}}_2(8){=3}}$$

$${{{power}}_2(32)=5}$$

$${{{power}}_3(9)=2}$$

$${{{power}}_3{(81)=4}}$$

$${{{power}}_5(25)=2}$$

Now, see if you can fill in the blanks:

$${{{power}}_2(16)=}$$ ____	$${{{power}}_{19}\left ( 1\over1000 \right )=}$$ ___
$${{{power}}_6(36)=}$$ ____	$${{power}}$$__$${(81)=2}$$
$${{power}}_5$$( ___ )$${=3}$$	$${{power}}$$___$${(81)=4}$$
$${{power}}_2$$( ___ )$${=-1}$$	$${{power}}_{16}$$( ___ )$${={3\over2}}$$

Guiding Questions

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References

f(t) Introducing Logs

Introducing Logs by Kate Nowak is made available on Function of Time under the CC BY-NC-SA 3.0 license. Accessed Feb. 22, 2018, 4:30 p.m..

Problem 3

What are possible value(s) for $$x$$ and $$y$$ that will make the statement true?

$$\mathrm{log}_x1=y$$

$$\mathrm{log}_xx=y$$

$$\mathrm{log}_x0=y$$

Guiding Questions

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Target Task

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Problem 1

Why is it necessary to specify that $${0<b<1}$$ and $${b>1}$$ are valid values for the base $$b$$ in the expression $$\mathrm{log}_b(x)$$?

References

EngageNY Mathematics Algebra II > Module 3 > Topic B > Lesson 8 — Exit Ticket, Question #1

Algebra II > Module 3 > Topic B > Lesson 8 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Modified by Fishtank Learning, Inc.

Problem 2

Evaluate the following logarithms:

$${\mathrm{log}_3(81)}$$

$${\mathrm{log}_{11}(1)}$$

$${\mathrm{log}_9(9)}$$

References

EngageNY Mathematics Algebra II > Module 3 > Topic B > Lesson 8 — Exit Ticket, Question #2

Modified by Fishtank Learning, Inc.

Additional Practice

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

Include problems translating between logarithm form and exponential form.
Ask for logarithms that evaluate to $$0$$ and $$1$$.
Include problems using variables, for instance, of the form $${\mathrm{log}_x8=3}$$ or $${\mathrm{log}_xx^4}$$.

EngageNY Mathematics Algebra II > Module 3 > Topic B > Lesson 8 — Exercises and Problem Set (Choose problems that evaluate to whole numbers)
EngageNY Mathematics Algebra II > Module 3 > Topic B > Lesson 8 — Examples/Non Examples section (Create error analysis based on these examples, emphasizing possible arguments for each component of the log)

Lesson 6

Lesson 8

Lesson Map

Topic A: Modeling with and Interpreting Exponential Functions

Identify, model, and analyze geometric sequences.

F.IF.A.3 F.IF.B.5 F.LE.A.2

Analyze and construct exponential functions that model contexts.

F.IF.B.4 F.IF.C.8.B F.LE.A.2

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

F.BF.A.1.A F.LE.B.5

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

A.SSE.B.3.C F.BF.A.1.A

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

A.SSE.B.4

Find the sum of an infinite geometric series.

A.SSE.B.4

Topic B: Definition and Meaning of Logarithms

Describe and evaluate simple numeric logarithms (Part I).

F.LE.A.4

Describe and evaluate simple numeric logarithms (Part II).

F.LE.A.4

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

F.BF.B.3 F.BF.B.4.B F.BF.B.4.C F.BF.B.5 F.IF.C.7.E

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

F.BF.B.4.C F.LE.A.4

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

F.LE.A.4

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

F.LE.A.4

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

F.BF.B.4.B F.LE.A.4

Solve equations with logarithms.

F.LE.A.4

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

A.SSE.A.1.B F.LE.A.4

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

A.SSE.A.1.B F.LE.A.4

Exponential Modeling and Logarithms

Objective

Common Core Standards

Core Standards

Linear, Quadratic, and Exponential Models

Foundational Standards

High School — Number and Quantity

High School — Number and Quantity

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

Problem 2

Guiding Questions

References

Problem 3

Guiding Questions

Target Task

Problem 1

References

Problem 2

References

Additional Practice

Lesson Map

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