Given $\log(p(x)) = q(x)$ are $p$ and $q$ algebraically independent?

Question

Since $e^x$ and $\log y$ are transcendental functions, does $$\log p(x) = q(x)$$ mean that polynomials $p$ and $q$ (of finite degree $n$ and $m$ respectively) are algebraically independent?

What happens if we allow $n,m\longrightarrow\infty$ ?

Answer 1

As a start, consider the following: If $p$ and $q$ are polynomials related via $e^{q(x)} = p(x)$, then $q$ must be a constant polynomial; this can be seen by noting that $e^x$ grows faster than any polynomial.

We also see that $p$ must also be constant.