solving in x involving both exponential and logarithmic function

Question

Is it possible to solve a function with both exponential and logarithm such as
$a x^2−b.\log(x)= c$
in closed form; where $a,b,c$ are constants and $a>0$ and $b>0$?

Answer 1

First it should be said that $x^2$ is not exponential, since $x$ is your independent variable. An exponential terms generally means something of the form $e^{kx}$. Your equation is transcendental and often does not have a solution in algebraic numbers (unlike quadratic eqations, for instance).

Answer 2

It depends what you mean by "solve". In general, not using elementary functions, no. The general solution to the problem you posted above involves Lambert's $W$-function.