# Engaging students: Solving exponential equations

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Jesse Faltys. Her topic: solving exponential equations.

APPLICATIONS: What interesting (i.e., uncontrived) word problems using this topic can your students do now?

Once your students have learned how to solve exponential equations, they can solve many different kinds of applied problems like population growth, bacterial decay, and even investment earning interest rate.    (Examples Found: http://www.education.com/study-help/article/pre-calculus-help-log-expo-applications/)

Examples

1. How long will it take for $1000 to grow to$1500 if it earns 8% annual interest, compounded monthly? $A = P \left( 1 + \displaystyle \frac{r}{n} \right)^{nt}$

• $A (t) = 1500$, $P = 1000$, $r = 0.08$, and $n = 12$.
• We do not know $t$.
• We will solve this equation for $t$ and will round up to the nearest month.
• In five years and one month, the investment will grow to about \$1500.

2. A school district estimates that its student population will grow about 5% per year for the next 15 years.  How long will it take the student population to grow from the current 8000 students to 12,000?

• We will solve for t in the equation $12,000 = 8000 e^{0.05t}$. $12,000 = 8000 e^{0.05t}$ $1.5 = e^{0.05t}$ $0.05t = \ln 1.5$ $t = \displaystyle \frac{\ln 1.5}{0.05} \approx 8.1$

• The population is expected to reach 12,000 in about 8 years.

3. At 2:00 a culture contained 3000 bacteria.  They are growing at the rate of 150% per hour.  When will there be 5400 bacteria in the culture?

• A growth rate of 150% per hour means that $r = 1.5$ and that $t$ is measured in hours. $5400 = 3000 e^{1.5t}$ $1.8 = e^{1.5t}$ $1.5t = \ln 1.8$ $t = \displaystyle \frac{\ln 1.8}{1.5} \approx 0.39$

• At about 2:24 ( $0.39 \times 60 = 23.4$ minutes) there will be 5400 bacteria.

A note from me: this last example is used in doctor’s offices all over the country. If a patient complains of a sore throat, a swab is applied to the back of the throat to extract a few bacteria. Bacteria are of course very small and cannot be seen. The bacteria are then swabbed to a petri dish and then placed into an incubator, where the bacteria grow overnight. The next morning, there are so many bacteria on the petri dish that they can be plainly seen. Furthermore, the shapes and clusters that are formed are used to determine what type of bacteria are present.

CURRICULUM — How does this topic extend what your students should have learned in previous courses?

The students need to have a good understanding of the properties of exponents and logarithms to be able to solve exponential equations.  By using properties of exponents, they should know that if both sides of the equations are powers of the same base then one could solve for x.  As we all know, not all exponential equations can be expressed in terms of a common base.  For these equations, properties of logarithms are used to derive a solution.  The students should have a good understanding of the relationship between logarithms and exponents.  Logs are the inverses of exponentials.  This understanding will allow the student to be able to solve real applications by converting back and forth between the exponent and log form.  That is why it is extremely important that a great review lesson is provided before jumping into solving exponential equations. The students will be in trouble if they have not successfully completed a lesson on these properties. TechnologyHow can technology be used to effectively engage students with this topic?

1. Khan Academy provides a video titled “Word Problem Solving – Exponential Growth and Decay” which shows an example of a radioactive substance decay rate. The instructor on the video goes through how to organize the information from the world problem, evaluate in a table, and then solve an exponential equation. For our listening learners, this reiterates to the student the steps in how to solve exponential equations.

2. Math warehouse is an amazing website that allows the students to interact by providing probing questions to make sure they are on the right train of thought.

For example, the question is $9^x = 27^2$ and the student must solve for $x$.  The first “hint” the website provides is “look at the bases.  Rewrite them as a common base” and then the website shows them the work.  The student will continue hitting the “next” button until all steps are complete. This is allowing the visual learners to see how to write out each step to successfully complete the problem.

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