  We construct a Hennings type logarithmic invariant for restricted quantum
$\mathfrak{sl}(2)$ at a $2\mathsf{p}$-th root of unity. This quantum group $U$
is not braided, but factorizable. The invariant is defined for a pair: a
3-manifold $M$ and a colored link $L$ inside $M$. The link $L$ is split into
two parts colored by central elements and by trace classes, or elements in the
$0^{\text{th}}$ Hochschild homology of $U$, respectively. The two main
ingredients of our construction are the universal invariant of a string link
with values in tensor powers of $U$, and the modified trace introduced by the
third author with his collaborators and computed on tensor powers of the
regular representation. Our invariant is a colored extension of the logarithmic
invariant constructed by Jun Murakami.
