Chi-square law from a transformed power-law variable

The probability density function of a random variable $X$ is given by

$$f(x)=\begin{cases}\alpha x^{\alpha-1}, & 0<x<1,\ 0, & \text{otherwise,}\end{cases}\quad\alpha>0.$$

Then the distribution of the random variable $Y=\log_eX^{-2\alpha}$ is