The Beautiful and Useful Applications of Logarithms

Proving a number theory result using logarithms

Logarithms are among the most useful tools we have at our disposal in mathematics. They allows us to translate problems of a multiplicative nature to problems of an additive nature, and this can often be the key to unlocking a solution to a problem. They are especially useful for problems involving powers or indices.

To illustrate this, let’s look at an interesting number theory problem. We are looking to find all positive integer pairs n and m which satisfy the equation n^m = m^n, when n and m are distinct. If we play around with small n and m we can quickly see that 2⁴ = 16 = 4², so the pair 2 and 4 are certainly one solution.

It turns out that the pair 2 and 4 is in fact the only such solution. The more interesting aspect of this problem is how we prove that this is the only solution. The proof makes great use of logarithms and reduces to an analysis of a function which is continuous in the positive domain, rather than a discrete number theory approach. Let’s do it.

Turning the problem into a function to analyze

If we do some simple manipulation of our equation through taking natural logarithms, we can state it in an alternate form, as follows: