Exponential Modeling and Logarithms

Lesson 15

Math

Unit 5

11th Grade

Lesson 15 of 16

Objective


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

Common Core Standards


Core Standards

  • 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.
  • 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

  • F.LE.A.1.A
  • F.LE.A.1.B
  • F.LE.A.1.C

Criteria for Success


  • Solve equations of the form $${a=bc^x}$$.
  • Construct exponential models.
  • Identify appropriate strategies to solve problems arising from exponential modeling situations.
  • Interpret solutions of exponential modeling problems in terms of a contextual situation.
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Anchor Problems


Problem 1

A population of a city grows at 5% per year. The population in 1990 was 400,000. In what year do you predict the population will reach one million?

Guiding Questions

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Problem 2

A bacteria doubles in size every six hours. Four hours from now, the culture has 8,000 bacteria. How many bacteria were there to start? 

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Problem 3

Algal blooms routinely threaten the health of the Chesapeake Bay. Phosphate compounds supply a rich source of nutrients for the algae, Prorocentrum minimum, responsible for particularly harmful spring blooms known as mahogany tides. These compounds are found in fertilizers used by farmers and find their way into the Bay with run-offs resulting from rainstorms. Favorable conditions result in rapid algae growth ranging anywhere from 0.144 to 2.885 cell divisions per day. Algae concentrations are measured and reported in terms of cells per milliliter (cells/ml). Concentrations in excess of 3,000 cells/ml constitute a bloom.

  1. Suppose that heavy spring rains followed by sunny days create conditions that support 1 cell division per day and that prior to the rains Prorocentrum minimum concentrations measured just 10 cells/ml. Write an equation for a function that models the relationship between the algae concentration and the number of days since the algae began to divide at the rate of 1 cell division per day.
  2. Assuming this rate of cell divison is sustained for 10 days, present the resulting algae concentrations over that period in a table. Did these conditions result in a bloom?
  3. If conditions support 2 cell divisions per day, when will these conditions result in a bloom? 
  4. Concentrations in excess of 200,000 cells/ml have been reported in the Bay. Assuming the same conditions as in (c), when will concentrations exceed 200,000 cells/ml?

Guiding Questions

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References

Illustrative Mathematics Algae Blooms

Algae Blooms, accessed on Feb. 23, 2018, 11:39 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


You deposit $8,000 into an account that pays 4.5% interest, compounded quarterly. How long will it take for you to have $12,000?

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.

  • Focus on analyzing the structure and meaning of the values and variables in an equation, generating an equation to find the model, and then solving the model. 
  • Include problems solving equations of the form $${a=bc^x}$$ separate from any context
  • Include problems involving exponential growth and decay, using interest and other contexts
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Lesson 14

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Lesson 16

Lesson Map

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Topic A: Modeling with and Interpreting Exponential Functions

Topic B: Definition and Meaning of Logarithms

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