Let G = (V, E) be a connected simple graph. A bijection f :holds for any two adjacent vertices u and v, where f + (u) = e∈E(u) f (e) and E(u) is the set of edges incident to u. A graph G is called local antimagic if G admits at least a local antimagic labeling. The local antimagic chromatic number, denoted χ la (G), is the minimum number of induced colors taken over local antimagic labelings of G. Let G and H be two disjoint graphs. The graph G[H] is obtained by the lexicographic product of G and H. In this paper, we obtain sufficient conditions for χ la (G[H]) ≤ χ la (G)χ la (H). Consequently, we give examples of G and H such that χ la (G[H]) = χ(G)χ(H), where χ(G) is the chromatic number of G. We conjecture that (i) there are infinitely many graphs G and H such that χ la (G[H]) = χ la (G)χ la (H) = χ(G)χ(H), and (ii) for k ≥ 1, χ la (G[H]) = χ(G)χ(H) if and only if χ(G)χ(H) = 2χ(H) + ⌈ χ(H) k ⌉, where 2k + 1 is the length of a shortest odd cycle in G.