Solve $x=C \log(C \log(x+A)+B)$

Solution 1:

This answer is for solving the equation by closed-form solutions. A closed-form function is a function from a given set of allowed functions.

The elementary functions according to Liouville and Ritt are those functions which are obtained in a finite number of steps by performing only algebraic operations, exponentials and/or logarithms. A Liouvillian function is an elementary function or (recursively) the integral of a Liouvillian function, e.g. the nonelementary integral of an elementary function.

Assume an ordinary equation $F(x)=c$ is given where $c$ is a constant and $F$ is a function. Isolating $x$ on one side of the equation only by operations to the whole equation means to apply a suitable partial inverse function $F^{-1}$ of $F$: $\ x=F^{-1}(c)$.

The problem of existence of elementary inverses of elementary functions is solved in Ritt, J. F.: Elementary functions and their inverses. Trans. Amer. Math. Soc. 27 (1925) (1) 68-90. Unfortunately your equation doesn't seem to be able to be brought into the form of an equation which is solvable through elementary functions.

The elementary functions form a differential field, and the Liouvillian functions form also a differential field. But you also can take other differential fields. The problem of solving a given equation by solutions from a differential field is solved in Rosenlicht, M.: Liouville's theorem in Differential Algebra. Publications Mathématiques de l'IHÉS. 36 (1969) 15-22. Unfortunately your equation doesn't seem to be able to be brought into the form of an equation which is solvable through elementary functions.

For applying only Lambert W and elementary functions, your equation should be in the form

$$f_1(f_2(x)e^{f_2(x)})=c,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$$

where $c$ is a constant and $f_1$ and $f_2$ are elementary functions with a suitable elementary partial inverse. Unfortunately your equation doesn't seem to be able to be brought into this form.