Find solutions of $\alpha x^n = \ln x$ How can I find the solutions of $$\alpha x^n = \ln x$$ when $\alpha \in \mathbb{R}$ and $n\in \mathbb{Q}$? Or, if it is not possible to have closed form solutions, how can I prove that there exist one (or there is no solution) and that it is unique? (I'm particularly interested in the cases $n=2$, $n=1$, and $n=1/2$).

In the case that $\alpha n\not =0$, we have
$$\alpha x^n = \ln x$$
$$ \alpha nx^{n}= n\ln x$$
$$ \alpha nx^{n}= \ln x^{n}$$
$$ e^{\alpha nx^{n}} = x^{n}$$
$$ 1= \frac{x^{n}}{e^{\alpha nx^{n}}}$$
$$ 1= x^{n}e^{-\alpha nx^{n}} $$
$$ -\alpha n= -\alpha nx^{n}e^{-\alpha nx^{n}} $$
$$ W(-\alpha n)= -\alpha nx^{n} $$
$$ -\frac{W(-\alpha n)}{\alpha n}= x^{n} $$
$$ x=\left(-\frac{W(-\alpha n)}{\alpha n}\right)^{\frac1n}$$
Where $W$ is the Lambert W function.
