How to get symmetry function of a function in proportion to any other functions generally?

Question

How can I get symmetry equation of a function in proportion to an other one? for instance we know that: $y=\log_{10}{x}$ and $y=\log_{\frac{1}{10}}{x}$ are symmetric in proportion of $y=0$. now, I want a general solution for achieving this desired(in ex. $y=0$) for all function.