Lesson 16 | Exponential Modeling and Logarithms | 11th Grade Mathematics

Objective

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

Common Core Standards

Core Standards

The core standards covered in this lesson

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.

High School — Algebra

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)<sup>n</sup> as the product of P and a factor not depending on P.

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<sup>ct</sup> = 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

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.

Criteria for Success

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

Solve equations of the form $${a=bc^{dx}}$$.
Solve problems using logarithm properties.

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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

Lincoln deposits money in a Certificate of Deposit account. The balance (in dollars) in his account t years after making the deposit is given by $${{{{L(t)}}}=500(1.05)^t}$$ for $${{t\geq0}}$$. Explain, in terms of the structure of the expression for $${{{L(t)}}}$$, why Lincoln's balance can never be $${499}$$.

Helen deposits money in a similar Certificate of Deposit account. The balance in her account is described, with units as in Lincoln's deposit, $${{{H(t)}}=600(1.04)^t}$$ for $${{t\geq0}}$$. Use the structure of the expressions for $${{{L(t)}}}$$ and $${{H(t)}}$$ to describe how the two balances compare over time.

By what percent does the value of $${{{L(t)}}}$$ grow each year? What about $${{H(t)}}$$? Explain.

During which year does Lincoln's balance to "catch up" with Helen's? Show your work.

Guiding Questions

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References

Illustrative Mathematics Comparing Exponentials

Comparing Exponentials, accessed on Feb. 23, 2018, 11:48 a.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Problem 2

A cup of hot coffee will, over time, cool down to room temperature. The principle of physics governing the process is Newton's Law of Cooling. Experiments with a covered cup of coffee show that the temperature (in degrees Fahrenheit) of the coffee can be modelled by the following equation

$${f(t)=110e^{−0.08t}+75}$$

Here the time $$t$$ is measured in minutes after the coffee was poured into the cup.

Explain, using the structure of the expression $$ 110e^{−0.08t}+75$$, why the coffee temperature decreases as time elapses.
What is the temperature of the coffee at the beginning of the experiment?
After how many minutes is the coffee 140 degrees? After how many minutes is the coffee 100 degrees?

Guiding Questions

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References

Illustrative Mathematics Newton's Law of Cooling

Newton's Law of Cooling, accessed on Feb. 23, 2018, 11:53 a.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Target Task

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

Potassium-40 has a half-life of $${1.28\times 10^9}$$ years, which means that after that many years, half of a sample has decayed into a different isotope of carbon. If a sample began with $${115}$$ grams of potassium-40 $${30,000,000}$$ years ago, how much is left today? How long will it be until there are only $${50}$$ grams remaining?

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.

The problems DDT-cay and Uranium 238 can be adapted to solve an equation of the form $${a=bc^{dx} }$$ to provide more practice on the skill in Anchor Problem #2.

Illustrative Mathematics Uranium 238
Illustrative Mathematics Carbon 14 Dating
Illustrative Mathematics DDT-cay
Illustrative Mathematics Bacteria Populations

Lesson 15

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

High School — Algebra

Linear, Quadratic, and Exponential Models

Foundational Standards

Linear, Quadratic, and Exponential Models

Linear, Quadratic, and Exponential Models

Linear, Quadratic, and Exponential Models

Criteria for Success

Anchor Problems

Problem 1

Guiding Questions

References

Problem 2

Guiding Questions

References

Target Task

Additional Practice

Lesson Map

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