logarithm function in a polynomial equation

Question

I faced the following polynomial equation which has a logarithm function inside of it: $\\ x^2 + \ln(x) - a = 0 $. I could manage to use approximation in small $x$, ($0<x \ll 1$) to solve this equation by the conventional methods ($\ln(1+x) \approx x$): $$ x = x^\prime + 1 \\ (x^\prime -1)^2 + x^\prime + a =0 $$ I wonder if I can solve this equation analytically without using such approximations. Thanks :)

Answer 1

Since $x>0$ $$x^2+\log(x)=a\implies x^2+\frac12 \log(x^2)=a$$ Let $x^2=t$, muliply by $2$ and exponentiate $$t+\frac12 \log(t)= a\implies 2t +\log(t)=2a\implies t e^{2t}=e^{2a}\implies 2t e^{2t}=2e^{2a}$$ Let $2t=u$ to make the nice $$ue^u=2e^{2a}\implies u=W\left(2 e^{2 a}\right)$$ where appears Lambert function.

Now, back to $t$ and $x$

$$x=\pm\sqrt{\frac{1}{2} W\left(2 e^{2 a}\right)}$$ which is defined $\forall a$.