I've got some real data that when plotted in log-log space is curved. See the image here. I've not labelled the axes as they're not important to this question.
So for this particular case there's two physical processes at work, and you can describe those with two relationships that intersect. I'm not asking for advice on curve fitting or describing this particular data.
The shape of the curve got me wondering if there is a mathematical relationship that is curved like this in log-log space? The motivation for this comes from a bit of curiosity, and I guess was partly motivated by the identification of power-laws and fractal dimensions in log-log space. I just wonder if there is a similar cool thing in maths that exists for curves in log-log space.
I have no formal mathematics training outside of what I did in my undergraduate degree in a STEM subject, so go easy. If you have direction to reading I'm more than happy to be told to go read a book/paper!
The fact that they're log-axes doesn't make fitting to data any harder. Let's say original variables $x,\,y$ have been transformed to $u:=\ln x,\,v:=\ln y$. Then your plot shows $v$ against $u$, although the values of the old variables written on the axes don't attest to this. If $v=f(u)$ is a good fit to what we see, it can be restated as $y=\exp f(\ln x)$. The choice of $f$ is the hard part, but then it always is. To take an example, one could propose a relation of the form $f=k/(u-c)+\ln A$, i.e. $y=A\exp(k/(\ln x-c))$.