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Exponential decay of gas prices: A logarithm and inverse case study

Exponential decay of gas prices: A logarithm and inverse case study

Author: Christopher Danielson

To support students working on an assignment dealing with exponential decay in a College Algebra class.

An exponential equation is solved by use of logarithms, and students are encouraged to consider the meanings of these equations and their solutions.

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Let's start with the following textbook problem:

To this problem, I have added the following tasks:

(c)    Based on the news report that $3.60 represents a 10% drop from the previous week, what was the price of gas one week before this report?
(d)    Part a asks you to assume that the relationship between time and the price of gas during this time period is exponential. In an exponential model, the decay factor (i.e. first ratio) will be constant over time. Assume instead that it the drop in price is linear. That is, assume that instead of a constant decay factor, we have a constant first difference. Create a linear model for this relationship.
(e)    Graph your linear and your exponential models on the same set of axes.

And I have challenged my students to go beyond the basic task by asking a question about this context that requires logarithms to answer precisely.

Source: Stewart, Redlin, Watson and Panman. (2011). College algebra: Concepts and contexts. Belmont, CA: Brooks/Cole.

A student's question

After class today, a student asked me about solving the following equation:

6 equals 3.60 times open parentheses 0.9 close parentheses to the power of x

The following video demonstrates the solution method.

Solving the equation

This video shows the symbolic solution to the equation.

Now let's step back for a moment

So now we have a solution. What we need to do is interpret the solution. Important questions to consider include these:

  1. What was the original equation asking about the context?
  2. What is the meaning of the solution?
  3. In particular, why is the solution negative?

My students are still working on this assignment for a grade, so I can't post my own answers to these questions here. But I do encourage them (and others) to post their own ideas and questions on the Q&A section for this packet.

If it turns out that the original equation wasn't asking what you meant to ask, maybe you need to write a new one and solve that.