  In this article, we estimate the essential norm of weighted composition
operator $W_{u, \varphi}$, acting on the logarithmic Bloch space
$\mathcal{B}^{v_{\log}}$, in terms of the $n$-power of the analytic function
$\varphi$ and the norm of the $n$-power of the identity function. Also, we
estimate the essential norm of the weighted composition operator from
$\mathcal{B}^{v_{\log}}$ into the growth space $H_{v_{\log}}^{\infty}$. As a
consequence of our result, we estimate the essential norm of the composition
operator $C_\varphi$ acting on the Logarithmic-Zygmund space.
