Significance of log in DCG metric?
From the wikipedia page:


  Previously there has not been any theoretically sound justification for using a logarithmic reduction factor2 other than the fact that it produces a smooth reduction. But Wang et al. (2013)[3] give theoretical guarantee for using the logarithmic reduction factor in NDCG. The authors show that for every pair of substantially different ranking functions, the NDCG can decide which one is better in a consistent manner.


The work showed that the log function is able to converge to consistent results for different ranking functions. It distinguishes the methods nicely even at high numbers of documents (at thelimit).

For example, here is a graph of the ranking measures using NDCG@k, where only the top k entries in the ranked list are considered: