It recently occurred to me that I did not know how to solve equations of the form $k^x=x^c$ for any two constants $k$ and $c$. After much pain in algebraically manipulating the equation (using logarithms) I confirmed the x is trapped in an exponent and cannot be extracted without some external knowledge.
Finding the roots of the derivative creates a similar problem.
I then did some research and discovered Lambert's W function, which I haven't the mathematical maturity to understand deeply. (Solve $2^x=x^2$).
I'm looking for some rigorous intuition towards why without numerical methods or Lambert's W function the equation is not solvable for x, or more specifically, a proof as to why this class of function is unsolvable using only logs and rationals? The best answer I have for myself is that because x is both a power and an exponent, either extracting x with a log or removing the exponent with a rational will keep the other side of the equation unsimplified. What is the mathematical language with which to express this idea?
As an aside, I'm also wondering what area of mathematics considers issues such as these (this one being on the simpler side).