= Solution
For the <Legendre transform>, maximize $px-f(x)$ over $x>0$. Its <derivative> is $p-1-\log x$, and strict concavity makes the stationary point $x=e^{p-1}$ the unique maximizer. Substitution gives the <convex conjugate of x log x>
$$
\boxed{f^*(p)=e^{p-1},\qquad p\in\mathbb R.}
$$
The range of $f'$ is all of $\mathbb R$, so all real dual arguments have this finite value. Every point of the original graph is supported by its tangent of slope $f'(x)$; the conjugate supremum therefore recovers $f$ from these supporting lines. Equivalently, the inverse <derivative> relation for this strictly convex differentiable <Legendre transform> gives \b[$f^{**}=f$ on $x>0$]. If an extended-real function on all of $\mathbb R$ is wanted, its closed convex extension has value zero at $x=0$ and infinity for $x<0$; biconjugation includes that endpoint closure.
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