Curriculum / Math / 11th Grade / Unit 5: Exponential Modeling and Logarithms / Lesson 16
Math
Unit 5
11th Grade
Lesson 16 of 16
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Lesson Notes
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Use logarithms to solve exponential modeling problems (Part II).
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)<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<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.
The foundational standards covered in this lesson
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 — Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
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.
The essential concepts students need to demonstrate or understand to achieve the lesson objective
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Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding
25-30 minutes
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.
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.
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.
A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved
5-10 minutes
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 $${115}$$Â grams of potassium-40 $${30,000,000}$$Â years ago, how much is left today? How long will it be until there are only $${50}$$ grams remaining?
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.
Topic A: Modeling with and Interpreting Exponential Functions
Identify, model, and analyze geometric sequences.
Standards
F.IF.A.3F.IF.B.5F.LE.A.2
Analyze and construct exponential functions that model contexts.
F.IF.B.4F.IF.C.8.BF.LE.A.2
Write and change the form of exponential functions that model compounding interest.
F.BF.A.1.AF.LE.B.5
Define and use $$e$$ in continuous compounding situations.
A.SSE.B.3.CF.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.
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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).
Describe logarithms as the inverse of exponential functions and graph logarithmic functions.
F.BF.B.3F.BF.B.4.BF.BF.B.4.CF.BF.B.5F.IF.C.7.E
Evaluate common and natural logs using tables, graphs, and calculators.
F.BF.B.4.CF.LE.A.4
Understand and apply the change of base property to evaluate logarithms.
Develop and use the product and quotient properties of logarithms to write equivalent expressions.
Develop and use the power property of logarithms to write equivalent expressions.
F.BF.B.4.BF.LE.A.4
Solve equations with logarithms.
Use logarithms to solve exponential modeling problems (Part I).
A.SSE.A.1.BF.LE.A.4
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