Throw a Few Logs on the Fire

When teaching apportionment and the Balinski-Young Impossibility Theorem to a group of gifted middle school students, it was necessary to discuss the geometric mean. Once the door was open, this was an opportunity to discuss logarithms, too. In particular, the geometric mean of two numbers is equal to the arithmetic mean of their logarithms.

As a trivial example of this property, the geometric mean of 10¹ and 10³ is 10², and the arithmetic mean of 1 and 3 is 2. But you probably saw that coming. It’s a bit more powerful to use less nice numbers. For instance, the geometric mean of 10¹ and 10² is approximately 31.62, and the arithmetic mean of 1 and 2 is 1.5. So what, right? Well, the kids in my class gasped audibly when their calculators revealed that 10^1.5 = 31.62.

One of the students even whispered, “Oh, man, that is SO cool.”

Though that wasn’t my best lesson, that was certainly the best reaction I ever received from a classroom of students. Quite honestly, it surprised me. But recently, cognitive scientists at MIT have theorized that it may be more natural for humans to think logarithmically than linearly, so perhaps there’s a genetic reason that kids think it’s cool. Apparently, both young kids and people from some tradditional societies will say that 3 is the number halfway between 1 and 9.

Of course, logrithms are great fodder for math jokes…

Lumberjacks make good mathematicians because of their logarithms.

Although some people think that a logarithm is a birth control method for foresters.

This, of course, is the natural log: ln(x). And this is an unnatural log…

But as we all know, a logarithm is nothing more than a misspelled algorithm.