Just wrapped up a unit on logarithms in Algebra 2 Honors. (p.s. It’s Friday and I’m officially on spring break!!!) [note: I didn’t finish this post when I started it, and now it’s Saturday and my break is officially almost over!!!!] In teaching logarithms for what is maybe the 8th year in a row, I managed to stumble on a different way to talk about the log properties that was like, okay, duh.
Let me preface this by saying I do think the way we talk about mathematics out loud can create or reinforce misconceptions for students about how math works. The best example I can point to is using “cancel” as an all-purpose term that students interpret as “these things disappear!” rather than taking the time and care to point out when inverse operations undo each other (“divides to one” or “adds to zero”). [See Nix the Tricks for more]
I feel the same way about sloppily using the word “distribute” when I’m not talking about the distributive property – especially given how quickly students make the mistake of trying to “distribute” operations. (i.e. log(X+Y) = log(X) + log(Y))
So, this year when I started talking about log properties, I was more careful about the way I spoke. After first noticing and then proving the properties by making the connection to exponent properties, I fell into a habit of describing them the same way I describe the exponent ones. That is, if we expanded log(XY) to log(X) + log(Y), I’d say “because when you multiply two powers, you add their exponents.” Ditto log(X/Y) = log(X) – log(Y) (“when you divide two powers, you subtract the exponents”) and log(A^C) = C*log(A) (“when you raise a power to a power, you multiply the exponents”).
Might seem like a small shift, but I caught myself a time or two reverting back to really imprecise language like “so the multiplication here becomes addition of the two logs” and realized how irritating and confusing this sounds. What’s happening with these log properties isn’t a magical swap of one operation for another, and these aren’t algebraic rules that students should feel like they are memorizing. Instead, these properties are a natural extension of ideas the students are really familiar and comfortable with – exponents.