Is it possible to resolve an equation of the type
$$x=C\log{(C\log{(x+A)}+B)}$$
(where $A$, $B$, and $C$ are real-valued parameters) for $x$?
As far as I can see, the function on the right hand side is concave for positive $C$ (for negative $C$ it may be partly convex, partly concave) and the equation has two real solutions if $C$ is large enough (consider $C=10$, $A=1$, $B=1$), otherwise none. (For negative $C$, the equation may have one real solution.)
My attempts to resolve the equation for $x$ were not very successful and Wolframalpha does not give a solution either (though it computes the numeric solutions for any specific parameter set). From what I gathered, even the solution of a similar function with a single logarithm (not the double one as above) involves the transformation into a form with Lambert's W function which is then impossibly difficult to work with (to further resolve the argument if you have another logarithm in there...) and which may even be impossible to derive if there is another logarithm in the intended argument (?).
Is there another possibility to solve the above equation or does this mean, that the solution is either currently or generally not possible? If so, is there scientific literature dealing with this problem (either saying why this or a similar problem is impossible to resolve or detailing what would have to be done in order to be able to solve it)?