“By shortening the labours [he has] doubled the life of the astronomer.” -Pierre-Simon Laplace

I had such an interesting conversation in my pre-service math class the other day. We were solving the equation

My goal here was to get them thinking about how they could us the power laws to help them. We worked our way down to

And someone offered the suggestions that the 3^8 and the 3^4 x 3^4 were the same, so all we really needed to determine was

and we eventually settled down on x = 2, since 4^2 = 16. Then I did something weird. I told them to pull out their calculator and evaluate

to which they found the answer to be 2. Now I had them intrigued. How were logarithms connected to this question?!

They knew that logarithms were a pre-calculus operation, but hadn’t made that connection between logarithms and exponentiation. It is likely that logarithms had been taught as a series of rules to follow, without a real explicit connection to how they are the inverse operation of exponentiation – or, more importantly (in my opinion) how they solve one piece of the “triple puzzle.”  You see, the process of exponentiation involves three values: the base (a), the power (p), and the evaluation (b).

We could cover any one of these numbers up, so we have three different but related problems. (1) We could cover up b; this problem can be solved by the process of exponentiation.

(2) We could cover up a; this problem can be solved by applying a radical.

(3) We could cover up p; this problem can be solved by applying a logarithm.

I am not convinced that students get enough time exploring  and developing their sense of logarithms, so I suggest utilising a structure that was brought up by David Butler today (the triangle typically used in science courses to remember arrangements of formulas). I do think in our case, the structure of the triangle works a bit better than it does for science formulas. Here, the triangle works nicely for a^p = b:

What if we fill in two values:

Is it possible that we can create meaning about logarithms by using these diagrams to introduce the three similar, but related, problems? I think so. We know 2^3 = 8, so how might we reason through this?

Well, we know ? must be close to 3 since if ? = 3, we would get 8. We also know ? must be larger than 2 since 2^2 = 4. Hmm… here we might begin to introduce the clunky notation of logarithms. Perhaps log(7)/log(2), or log 2(7). Aha! 2.807 seems reasonable based on what we have thought about. And we can now flush out the problem of non-integer powers.

Anyway, I don’t think we will ever be able to get rid of the unfortunate notational issues with logarithms, but I do think we can do better making the connections back to exponentiation. Maybe there is some space in the progression of learning about exponentiation for triforce notation? As always, I welcome your thoughts.

1. It is likely that logarithms had been taught as a series of rules to follow, without a real explicit connection to how they are the inverse operation of exponentiation.

I think you are likely being unfair on the original teachers. I would bet that they taught quite explicitly that logs are the inverse of exponents — that their students couldn’t retain it through the years doesn’t mean it wasn’t taught. I know that in my specific case I teach it explicitly and repeatedly, but many of my students just don’t get it.

Logs are incredibly difficult to teach well, and were one of the topics that it took me longest to come to terms with teaching. The first time I taught it I recall doing so very badly indeed.. There’s a lot going on, and if the students arrive a bit unsure of what an exponent is and does then you are already starting on the back foot.

By the same token, if a pre-service teacher finds the log/exponent relationship hard, then perhaps they should not be teaching very senior level Maths. You’ve got no show of teaching it well if your own understanding is not implicit.

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2. Interestingly enough, I spent more time last semester exploring and discussing the connection between logarithms and exponentiation. A student was using the triangle to reinforce the ideas.

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3. What a fun insight! I also recently stumbled across this idea of representing powers graphically in a triangle in a video from 3blue1brown: https://www.youtube.com/watch?v=sULa9Lc4pck

While I think the ending is a bit much (trying to actually use the triangle as notation when solving regular problems) I really liked the investigation aspect of it. I am thinking of trying it next time I teach the topic.

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