= Product of independent uniform random variables
{title2=$f_{XY}(z)=b^{-2}\log(b^2/z),\quad0<z<b^2$}
For <independent random variables> $X,Y$ with the <uniform distribution> on $(0,b)$, use the <probability density of a product of random variables> and integrate $1/(b^2x)$ over $z/b<x<b$. The displayed <probability density function> is zero off $(0,b^2)$ and has <integral> one. More generally, for <independent and identically distributed> factors $U_j$ with this <uniform distribution>, $W_n=\prod_{j=1}^nU_j$ satisfies
$$
\mathbb E[W_n^\alpha]=\left(\frac{b^\alpha}{1+\alpha}\right)^n\quad(\alpha>-1),\qquad \mathbb E W_n=(b/2)^n,\qquad\operatorname{Var}(W_n)=(b^2/3)^n-(b^2/4)^n.
$$
The <expectation of a product of independent random variables> proves these identities directly.
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