I was refreshing my memory on the power, exponential, nth-root, and logarithmic functions when I realised that there is a whole pair of functions missing.
If I know the exponent, I can make $y=x^a$ or its inverse $y=\sqrt[a]{x}$. If I know the base I can make $y=a^x$ and its inverse $y=\log_a(x)$. But what if I know the power and I want to define a relationship between the exponent and the base? It wasn't too hard to figure out that they should be $y=\log_x(a)$ and $y=\sqrt[x]{a}$. But I don't think I ever ran across these back when I was in school, and it sparked a lot of questions.
Do these functions have names? What (if anything) are they used for? Is there a reason why we don't teach them alongside those other related functions?
To my knowledge, those functions do not have their own name. The reason they are not taught about specifically is that they are less useful (in that do not appear as naturally in calculations as others) and more importantly, that they can easily be written in terms of known functions (namely the exponential and the natural logarithm) : $$f(x) = \sqrt[x]{a} = \exp\left(\frac{\ln(a)}{x}\right)$$ and $$g(x) = \log_x(a) = \frac{\ln(a)}{\ln(x)}$$