It is well known how to invert the following relation to get $y$ in terms of $x$
$$x=a\log(y-b),\quad\to\quad y=b+\exp(x/a)$$
Is there anything that can be said about inverting the relation
$$x=a\log(y-b)+c\log(y-d)$$
For general values of $a$, $b$, $c$, and $d$?
When $a=-c$, this is straightforward to invert
$$y=\frac{b+d\exp(x/a)}{\exp(x/a)-1}.$$
Also when $a=c$, we find
$$y=\frac{b+d\pm\sqrt{(b-d)^2+4\exp(x/a)}}{2}.$$
I'd like to understand how to invert the double logarithm relation in general. Perhaps there is a well known special function which allows one to write the general inversion formula?
Update
Taking the advice of Dasherman, we find
$$\exp(x)=(y-b)^a(y-d)^c.$$
This looks difficult in general to invert. The other special case I neglected is when $b=d$,
$$y=b+\exp(x/(a+c))$$