What other types of distributivity are there? When I say ‘Distributivity,’ I mean the way a number $x$ can be ‘Put in to’ some other function or the like.
For example, to distribute $x$ into $\text{id}_y$, you simply have to multiply $y$ by $x$ to get $\text{id}_{xy}$ (normal distribution). At first, I thought that this was the end for distribution. Until I found the $\log$ function.
I have found that, to distribute $x$ into $\log_b(y)$, all one has to do is to exponentiate $x$ onto $y$, as in, $\log_b(y^x)$.
I was wondering if there were other functions such that $x\cdot f(y)$ was equal to:


*

*$f(y+x)$

*$f(y-x)$/$f(x-y)$

*$f(x\div y)$/$f(y\div x)$

*etc.


(We already have examples for $f(xy)$ and $f(y^x)$, e.g.)

There is no real function such that $f(x+y) = xf(y)$
If this was true, you would have for all $x$, 
$$x^2f(y) = xf(x+y) = f(x+x+y) = 2xf(y)$$
