A Maths question that occupied us for the whole of sixth form

I’m worried that I might sometimes write a bit too seriously, so here’s something that’s a bit more light hearted… a maths problem.

There was just two of us in my Further Maths class, and we came up with this question and were obsessed with it for basically the whole two years. I used to walk around with a bit of paper with the question on inside my blazer pocket everyday, just in case we stumbled across another person we could ask. I was really cool back then I know. We asked everyone; anyone who was any good at Maths in the sixth form, all the Maths and Science teachers, yet no-one was able to answer it. It wasn’t until a few years ago that I remembered the question, attempted to solve it and managed to without too much hassle, to my shock.

If you know what exponential and log functions are, then continue to read as I’ll describe the problem now. If you don’t, then you might struggle with this next bit.

Imagine a curve of the form y equals a to the power of x, for a given value of a (that is greater than 1). Now on the same graph, imagine another curve of the form y equals log base a of x. If your imagination is struggling, then see this for some inspiration. Now the question was, what is the value of a such that the two curves intersect exactly once? Answers in the comments please.

Not quite Fermat’s Last Theorem, but took me almost as long to solve.