Question

Solve:
f(x,y)*f(y,x)=f(x,y)+f(y,x)

Answer 1

I promise to share the solutions I've found if you try. And my answers aren't trivial, they're dependent on x and y.

Answer 2

:D

Answer 3

So here's the answer I found:

\[f(x,y)=1+\log_xy\]

I honestly have no clue how I would find this sort of thing naturally, it came up when I was considering some interesting properties of logarithms, such as:

\[\log_a b \log_b a = 1\]
\[\log_a b \log_b c \log_ca= 1\]

Specifically I've just been looking for something so that

\[f(a,b) * \log_a b = \delta_{ab}\]
since this could be used to factor any number possibly easily with linear algebra.