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 2m = 8 and 3n = 9, what is the value of m · n?
This next one looks similar, but it’s a bit more difficult.
If 2m = 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 2m = 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 2m = 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 2m = p and 3n = q?
If 2m = 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 2m = 9, then 2mn = 9n = 32n = 82 = 26, so mn = 6.
My colleague Al Goetz solved it as follows.