Let $f(x)=g(x)$ be an equation (1) where at least one of $f$ and $g$ are transcendental functions. Let $h(x)=f(x)-g(x)$. If it can be shown that $h^{-1}(0)$ is non-algebraic, that implies that there is no algebraic solution to (1). How exactly does one go about doing this?
The answer here shows that there is no general method, but presumably there's a way to do this for individual equations - say, $xe^x=k, k\in\mathbb{R}$ (2). Obviously, the solution for, say, $k=3e^3$ is the algebraic value $x=3$, but this isn't an algebraic solution - I get it from sight. So in this sense, I'm using the term 'algebriac solution' to describe arriving explicitly at the form of $x=\text{constant}$ via some combination of arithmetic operations, as opposed to merely stumbling upon an algebraic number that satisfies an equation. Do we get a computer to run through every possible combination of operations on (2) to get that no solution exists, or is there a more sophisticated method available?
If such a method can't be generalised to all transcendental functions, can it be generalised to equations between specific families of functions? Like, $k^x=P_n(x)$ (where $k \in \mathbb{R}^{+}\backslash\{1\}$ and $P_n$ is a polynomial of degree $n>0$), doesn't have an algebraic solution. For example, it seems as though $2^x=3x^2-1$ is only solvable numerically (again, in spite of the solution $x=1$). How do we know this?
There's a very real possibility that I've abused mathematical vocabulary throughout. Apologies for length and/or if this has already been answered. If the answer is that it involves maths beyond what I've likely already been exposed to, could you point me in the right direction at least?
