A solution of an equation is elementary iff it is an elementary number.
Polynomial functions, rational functions and irrational algebraic functions of one variable are algebraic elementary functions. Their solutions are algebraic numbers. An algebraic number is elementary iff it can be represented as a radical expression. The problem when an equation of algebraic functions is solvable by radicals is solved by Galois theory. You are asking for solvability of equations of transcendental elementary functions in terms of elementary expressions.
This part answers when it is possible to find an elementary solution only by transforming the equation only by elementary operations that are readable from the equation.
This task can be related to the question of existence of closed-form inverses of the functions which are contained in the equation: If only elementary expressions are allowed, an equation $F(x)=0$ with elementary function $F$ given in closed form can be transformed according to $x$ simply by reading off the partial inverses of $F$ from the equation only if all partial inverses needed are elementary functions.
The incomprehensibly unfortunately hardly noticed theorem of Joseph Fels Ritt in Ritt, J. F.: Elementary functions and their inverses. Trans. Amer. Math. Soc. 27 (1925) (1) 68-90 answers which kinds of Elementary functions can have an inverse which is an Elementary function.
Risch also proved this theorem in Risch, R. H.: Algebraic Properties of the Elementary Functions of Analysis. Amer. J. Math 101 (1979) (4) 743-759 by his structure theorem for Elementary functions.
A method of proof for certain transcendental equations is given in Rosenlicht, M.: On the explicit solvability of certain transcendental equations. Publications mathématiques de l'IHÉS 36 (1969) 15-22.
Two methods for simpler transcendental elementary equations are given in [Lin 1983] Ferng-Ching Lin: Schanuel's Conjecture Implies Ritt's Conjectures. Chin. J. Math. 11 (1983) (1) 41-50 and in [Chow 1999] Chow, T.: What is a closed-form number. Am. Math. Monthly 106 (1999) (5) 440-448. Both need the proof of Schanuel's conjecture what currently is an unsolved mathematical problem.