# What ARE logarithms, though?

After their exponential growth and decay studies, my Math 3 class moved onto logarithms, a natural progression. Watching ViHart’s How I feel About Logarithms video inspired me, possibly more so than some of her others, so I had them watch it as an introduction. I love that her explanation compares logarithms (impossible!) to basic counting (easy!), along the way making points about the flawed way math is traditionally taught in classrooms. Watching this video, checking for understanding, and clarifying the meaning of logarithms takes a full class period, but I see this as worth it.

I created a companion handout for this film where students are asked to use the logarithmic scale to answer questions, and to interpret the different parts of a logarithmic expression. In their feedback to me this week, most said they had no trouble with the concept of logarithmic scale, but were confused about converting from log to exponent. The most surprising feedback that I received from multiple students is that this math seemed different from the things we had studied previously. To me, logarithms are such a natural sequel to exponentials because each function family is just asking a different question about the same topic. Clearly that connection is the part that is getting lost.

I should pause to say that it’s not just on this week with the complicated-seeming logarithms that I seek feedback. I have the students write a weekly journal, asking them to let me know how their week went in the class, then asking a follow-up about the math. I stole this idea from a teacher I had in high school, and encourage you to steal it from me! This week, they are really shaping my plans moving forward. On Monday we’ll focus on connecting logarithms to exponentials. They have already learned about inverses, and they have explored the concept of function families, so I’ll rely heavily on their prior knowledge.

Last week, just in case the logarithmic scale concept hadn’t been driven home quite yet, we did an activity designed to show logarithms’ connection to real life. I handed each student three slips of paper, each listing some length. These ranged from very large, like the distance from the earth to the moon, to very small, like the radius of a red blood cell. I asked them to put their slips in order, biggest to smallest. Then, I asked them to try to combine their list with a partner’s. This is not always immediately obvious – I wouldn’t have known that the Amazon River was longer than the diameter of the moon. Secretly, I wrote the size of these items in meters (all found here, I should point out) on the back. They flipped the slips over, converted these numbers into scientific notation, then placed them on this giant outdoor log-10 scale!

I’m totally using this again, especially if it’s as beautiful outside that day. I’m resolving to go outside more.