  In this paper, a class of $\log$-polyharmonic mappings
$\mathcal{L}_p\mathcal{H}$ together with its subclass
$\mathcal{L}_p\mathcal{H}(G)$ in the unit disk $\mathbb{D}=\{z: |z|<1\}$ is
introduced, and several geometrical properties such as the starlikeness,
convexity and univalence are investigated. In particular, we consider the
Goodman-Saff conjecture and prove that the conjecture is true in
$\mathcal{L}_p\mathcal{H}(G)$.
