Why the reduction from MINIMUM SET COVER to MINIMUM DOMINATING SET means $c \log n$-inapproximability for MINIMUM DOMINATING SET
I didn't at first understand that the question Inapproximability of set cover: can I assume m=poly(n)? contains the answer to my own question. (Thanks to @Tsuyoshi Ito for pointing out.) 

In short, the proof by Raz and Safra (and any other PCP-based proof) builds an instance of SET COVER where the number of subsets is a polynomial of the universe cardinality.