  We characterize the analytic self-maps $\phi$ of the unit disk ${\Bbb D}$ in
${\Bbb C}$ that induce continuous composition operators $C_\phi$ from the
log-Bloch space $\mathcal{B}^{\log}({\Bbb D})$ to $\mu$-Bloch spaces ${\mathcal
B}^\mu({\Bbb D})$ in terms of the sequence of quotients of the $\mu$-Bloch
semi-norm of the $n$th power of $\phi$ and the log-Bloch semi-norm (norm) of
the $n$th power $F_n$ of the identity function on ${\Bbb D}$, where $\mu:{\Bbb
D}\rightarrow (0,\infty)$ is continuous and bounded. We also obtain an
expression that is equivalent to the essential norm of $C_\phi$ between these
spaces, thus characterizing $\phi$ such that $C_\phi$ is compact. After finding
a pairwise norm equivalent family of log-Bloch type spaces that are defined on
the unit ball ${\Bbb B}_n$ of ${\Bbb C}^n$ and include the log-Bloch space, we
obtain an extension of our boundedness/compactness/essential norm results for
$C_\phi$ acting on ${\mathcal B}^{\log}$ to the case when $C_\phi$ acts on
these more general log-Bloch-type spaces.
