Take $-\log$ of a $\beta$ distributed R.V.

Question

$X_1,\dots,X_n \sim \beta(a,1)$, where $Y = -\log(X)$ Use the transformation formula to calculate the pdf of $Y$. What named distribution does it have?

I am confused what method to use here. A beta does not converge to a normal, so I cannot use the delta method?

Answer 1

Using the transformation formula, the density for $Y$ should be $$ f_Y(y) = e^yf_X(e^{-y}) = e^{-y}\frac{1}{B(a,1)}(e^{-y})^{a-1} = ae^{-ay}, \ \ \ 0 \leq y < \infty . $$ This is an exponential distribution