  We first provide a detailed proof of Kato's classification theorem of log
$p$-divisible groups over a noetherian henselian local ring. Exploring Kato's
idea further, we then define the notion of a standard extension of a classical
finite \'etale group scheme (resp. classical \'etale $p$-divisible group) by a
classical finite flat group scheme (resp. classical $p$-divisible group) in the
category of finite Kummer flat group log schemes (resp. log $p$-divisible
groups), with respect to a given chart on the base. These results are then used
to prove that log $p$-divisible groups are formally log smooth. We then study
the finite Kummer flat group log schemes
$T_n(\mathbf{M}):=H^{-1}(\mathbf{M}\otimes_{\mathbb{Z}}^L\mathbb{Z}/n\mathbb{Z})$
(resp. the log $p$-divisible group $\mathbf{M}[p^{\infty}]$) of a log 1-motive
$\mathbf{M}$ over an fs log scheme and show that they are \'etale locally
standard extensions. Lastly, we give a proof of the Serre-Tate theorem for log
abelian varieties with constant degeneration.
