Let G=(V,E) be a graph of order n without isolated vertices. A bijection f:V -- {1,2,...n} is called a local distance antimagic labeling if the weights of any two adjacent vertices are not equal, where the weight of a vertex is defined to be the sum of labels of adjacent vertices. The local distance antimagic chromatic number is defined to be the minimum number of colors taken over all colorings of G induced by local distance antimagic labelings of G. In this paper, we study the local distance antimagic chromatic number for the join of graphs and the lexicographic product of graphs with empty graphs.