Like yesterday’s lesson, we are going to use a context that will help us think about exponential and logarithmic functions. Today’s focus will be more on logarithms than on exponentials. To launch, have students watch the video about Yucca Mountain in Nevada. I heard about this nuclear waste repository in the book About a Mountain by John D’Agata (which I got for only $1 at Dollar Tree!) and it got me thinking about the math behind this.
After watching the video ask students the following questions to transition into the math of the lesson:
1. What is nuclear waste? (It is radioactive waste from nuclear power generation or a byproduct of medical or research endeavors that is hazardous to living things.)
2. What is a nuclear waste repository?
2. Why is it needed?
3. What happens to nuclear waste? (If students do not know, you can tell them that is decays exponentially.)
4. What does it mean to decay exponentially?
5. What other things decay exponentially? (Again, if students do not know you can tell them that isotopes do. Recall that isotopes are variations of elements that contain a different number of neutrons. This is used in carbon dating, which students may have heard about.)
Give students the task worksheet and have them work on the problems with their table. I would give students about 12 minutes to work through the two problems. Before letting them work, you may want to review what a half-life is and give a quick example.
Question #1 is pretty straightforward and is similar to what we worked on in yesterday’s lesson. However, some students may only estimate their answer. Since 46 years is just a little more than 3 half-lives, students may just calculate 3 half-lives and say that the answer is slightly less than 12.5%. If a student does this, ask them to find the exact amount. If they get stuck, you can ask them to find exactly how many half-lives will occur in 46 years. This will get them thinking about portions of half-lives.
For the second question, students might get a little lost. If they do, ask them to set up an expression to find out how much of the substance is left after x half-lives. This will get them thinking about setting up an exponential expression and they will see that the exponent is missing. My students usually have enough experience with logs to know that one is needed to find the missing exponent. Students may use guess and check to arrive at their answer. That is fine; when we have the class discussion we will remind them about logs.
Students must use the property logab = log b/log a to solve this problem. If there do not remember this from Algebra 2, we will review it in our class discussion.
Choose a student to show their solution to the first question. If possible, choose a student who just estimated the amount but did not find the exact amount. Then, ask a student who found the exact amount to explain how they got their solution. Stress that the exponent in this problem is just the number of half-lives, and to find the number of half-lives you just divide the number of year by the half-life.
When going over the second question, start with a student who could set up the exponential equation .08 = (.5)^t but could not solve for t. Ask students what the t stands for. Many students think it will represent the number of years instead of the number of half-lives. Then ask a student to show how their came to their solution. This is a good time to remind students about the property logab = log b/log a. My focus in precalculus is on the problem solving, not proving every last logarithmic property. The proof was hopefully completed in Algebra 2; we only have a certain amount of time and we cannot re-prove everything they’ve ever learned, so this is one that I choose to leave to their prior knowledge (more on that in this video). However, I always tell students that the proof is in their textbook and they can refresh their memory if they would like.
After getting the correct answer, ask how you could modify the equation (.08 = (.5)^t) to have the answer be the number of years to get to the safe level, rather than the number half-lives. Here are two different ways that my students set up the equation.
After going through the correct answer, start a discussion about logs in general. Ask students to write down five things that they remember about logarithms. In my experience, most students will write down properties instead of explanations of the meaning of a logarithm. Make sure they understand that logarithms are useful when the exponent is an unknown quantity. Highlight the fact that logarithmic functions are inverses for exponential functions. Students should also remember that an exponential equation such as 2^x = 8 can be written in logarithmic form as x = log28.
This assignment will give them some basic questions about logarithms to refresh their memory and is a nice summary of the concepts we worked on today.