A Small Change: Solving Equations with Logarithms

In my Math 10 class, did my lesson today involving solving exponential equations that cannot be solved using knowledge of integral powers. My start was the same as it has been for that lesson over many years:

I have students start with an iterative guess-and-check method since it’s something that will pretty much always work. This was no big deal to the students. When one student said her TI calculator gave the exact answer, I asked if she really thought that was the exact answer. She said no, but I used Python to rub it in a bit.

This was another opportunity to show the difference between exact and approximate answers – always something I try to teach implicitly whenever it comes up. As with many of the Common Core Standards for Mathematical Practice, I think this (MP6 – Attend to Precision) is always an idea that comes with context.

The big shift in this lesson came when we started solving the equation algebraically. I always do a bit of hand-waving at this point saying ‘isn’t it great that these logarithm properties let us do this?’, while getting a class full of students giving me just enough of a sarcastic head nod to make me feel bad about it.

Instead, I made reference to the process of switching back and forth from logarithmic and exponential form.

The students are pretty skilled at doing this. I wrote it up in the notes myself because most students wrote it faster than I could get anyone to explain the process.

The key here was that when I asked students to calculate these values on the calculator, nobody could do it. One found the LOGBASE command on their TI, but for the most part, this stayed as an abstract number. It made sense to them that they ended up with ‘x =’ in the end, but that didn’t make a big difference in terms of being able to talk about what that meant. They did a couple of these on their own.

Only then did I show them the logarithm property trick that lets us get the answer in a different form:

I admittedly connected some dots here, but I didn’t do so in a formal way of introducing change of base. A couple of them figured out that this was a form that they could calculate using the common logarithm button on their calculators.

I’m not emphasizing log properties this year outside of what they allow us to do in solving equations. This is something that we will devote more time to next year in IB Mathematics year 1 class. I will mention this change of base property as a nice tool to use for confirming graphical and iterative solutions, but probably won’t assess them knowing how to apply change of base directly.

Any time I can get rid of hand-waving and showing mathematics as a list of tricks to be memorized, it’s a win.