Is there such thing as this?

 First off, I want to point out something:

1. I know this is almost always used for help on certain questions, but I'm asking this question here anyways.

All right. Now that this is out of the way, I want to ask this question:

I know there is such thing as a fractional derivative, but is there anything like that that applies to logarithms? I'm not talking about a logarithm with a fraction for a base. Fractional derivatives are basically taking a certain number $n$ derivatives, where $n$ is not an integer, of a certain function $f(x)$ (An example would be $\frac{1}{2}$ derivatives of $\sqrt{x}$. This equals approximately 1.535348 for those who are wondering). What I am asking is is there a way to take, for example, a half a logarithm of something?

Thanks in advance and sorry if it seems confusing to you.