  The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the
least integer $d$ such that $G$ has an vertex labeling (edge labeling) with $d$
labels that is preserved only by a trivial automorphism. The lexicographic
product of two graphs $G$ and $H$, $G[H]$ can be obtained from $G$ by
substituting a copy $H_u$ of $H$ for every vertex $u$ of $G$ and then joining
all vertices of $H_u$ with all vertices of $H_v$ if $uv\in E(G)$. In this paper
we obtain some sharp bounds for the distinguishing number and the
distinguishing index of lexicographic product of two graphs. As consequences,
we prove that if $G$ is a connected graph with a special condition on
automorphism group of $G[G]$ and $D(G)> 1$, then for every natural $k$,
$D(G)\leq D(G^k)\leq D(G)+k-1$, where $G^k=G[G[...]]$. Also we prove that all
lexicographic powers of $G$, $G^k$ ($k\geq 2$) can be distinguished by at most
two edge labels.
