Interesting Exponent Problem

In the expression 3x², the 2 is called an exponent. And in the expression 4y³, the 3 is called the y‑ponent.

Speaking of exponents, here’s a problem you’ll have no trouble solving.

If 2^m = 8 and 3ⁿ = 9, what is the value of m · n?

This next one looks similar, but it’s a bit more difficult.

If 2^m = 9 and 3ⁿ = 8, what is the value of m · n?

One way to solve this problem is to compute the values of m and n using a logarithm calculator, though there’s an algebraic approach that’s far more elegant. (I’ll provide the algebraic solution at the end of this post, so as not to spoil your fun.)

Take a few minutes to solve that problem. Go ahead, give it a try. I’ll wait for you.

[…]

Ah, good. Glad you’re back.

Are you surprised by the result? I sure was.

It made me wonder, “If the product of the two numbers is 72, will it always be the case that m · n = 6?”

It only took me a minute to realize that the answer is no, which can be seen with numerous counterexamples. For instance, if 2^m = 64, then m = 6. This would mean that 3ⁿ = 9/8, because 64 × 9/8 = 72, but it would also mean that n = 1, since m · n = 6. That is obviously a contradiction, since 3¹ ≠ 9/8.

If the product of the two numbers is held constant at 72, then the problem looks something like this:

If 2^m = k and 3ⁿ = 72/k, what is the value of m · n?

The following graph shows the result for 1 < k < 72.

So that leads to a couple of follow-up questions.

For what integer values of p, q will p · q = 72 and m · n = 6, if 2^m = p and 3ⁿ = q?

If 2^m = p and 3ⁿ = q, for what integer values of p, q will m · n = 6 when m, n are not integers?

The answers, of course, are left as an exercise for the reader.

---

Here’s the algebraic solution to the problem above.

Since 2^m = 9, then 2^mn = 9ⁿ = 3²ⁿ = 8² = 2⁶, so mn = 6.

My colleague Al Goetz solved it as follows.

Since

Then

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.