I'm wondering if there exists a closed-form or analytic expression for the roots of an equation of the form
$ax^2 + bx + c\log x=0.$
considering the natural $\log$. Wolfram alpha is leading me to expressions involving the Lambert W (product log) function when I include either the quadratic term or the linear term (but not both) and analytic approximations when I supply real values for the coefficients.
This is OK, but does a more general solution exist in terms of the coefficients?
thanks!
Best Answer
Any equation which can write $$A+B x+C \log(D+Ex)=0$$ has a solution which can be expressed in terms of Lambert function. $$x=\frac{C }{B}W(F)-\frac{D}{E} $$ where $$F=\frac{B }{C E} e^{\frac{B D-A E}{C E}}$$This is why Wolfram Alpha gave you solutions for $$ax^2 + bx + c\log x=0$$ in the case where $a=0$ or $b=0$.
However, for the general case you are considering $(a \neq 0,b\neq 0)$, I do not think that exists any analytical solution and only numerical methods (such as Newton) would solve the problem and this should be quite easy.