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

Composition operators from logarithmic Bloch spaces to weighted Bloch spaces