Density of a product of seven uniforms

Let $X_1,\dots,X_7$ be independent $\text{Unif}(0,1)$ variables and $Y=X_1X_2\cdots X_7$. The density of $Y$ has the form $f(y)=\dfrac{(-\ln y)^a}{b}$ for $0<y<1$ (and $0$ elsewhere), with integers $a$ and $b$. Find $a+b$.

$726$. Each $-\ln X_i$ is $\text{Exp}(1)$, so $-\ln Y$ is a sum of seven of them, i.e. $\text{Gamma}(7,1)$. Changing variables back with $y=e^{-g}$ gives $f(y)=\tfrac{(-\ln y)^6}{6!}$ on $(0,1)$, so $a=6$, $b=720$ and $a+b=726$.