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When I was in high school, I studied logarithms, and learned a lot of their properties. The product rule, the quotient rule, the power rule, and the change of base rule were all properties I got very familiar with. Truth is, though, that as extensive as my math training was, there was a log property that my high school curriculum didn't mention. It was the base swapping rule*. I tell my students this as a way of pointing out that it's unwise to ever think you know everything.

After that discussion, I like to lead my students into discovering a property that isn't in their textbook. It's this one:

log_a^b m =

I don't show them the property, and I don't tell them that I'm going to introduce a new property. I start off by giving them a series of logs to evaluate. The logs come in pairs, with the second log in each pair having its base the square of the base in the first log. For example:

log₂ 16 and log₄ 16
log₃ 9 and log₉ 9
log₅ 625 and log₂₅ 625

In each case, it's clear that the second log is half of the first one. I ask students to explain the pattern, and someone will observe that the second base is the square of the first, and the log is half of the first. I then suggest that we try another example to test out our theory.

log₃ 81 and log₉ 81

Now I suggest that we might want to try to generalize this conclusion a bit. We start with a general log equation and manipulate it:

x = log_a^b m
(a^b)^x = m
a^bx = m
log_a m = bx
= x

Then, since x also equals log_a^b m, we can write:

log_a^b m =

log_a m

b

Since the rule is not in their textbooks, I tell them that we get to name the property. They may come up with a silly name, or they may decide to name it after the teacher, or the school, or something else altogether. But whatever you name the property, students will love it whenever they get to use that property, because they get to use the name they came up with for it.

* The base swapping rule (not the base changing rule) doesn't show up in all logarithm curricula. The property states:

log_a b =

This can be readily proven as well - this requires one application of the base changing rule:

x = log_a b
x =
x =

Lesson by Mr. Twitchell