Here's an idea. Take the scaled function: g(x, y; k) = log( exp(kx) + exp(ky) ) / k

Let k be 1/max(x,y) so that you have:

g(x,y) = log(exp(1)+exp(min(x,y)/max(x,y))*max(x,y)

There are some issues if the maximum value can be negative or zero, though.