@Rahul Sharma, If asymptotic calculations give some answer and in actual runtime it gives opposite result, then there's no point in studying algorithms. It has to be somewhat near to exact mathematical result.
Anyway, let's take some examples of complete graph.
1. V=128, E(max) =8128
2. V=1024, E=523776
We can take many such examples that will prove LogV is always less than LogE.
Other than trees, for a connected graph it is always likely that no.of edges are greater than equals to no.of vertices. It is somewhat V <= E < V^2.
So it should be LogV = O(LogE)
I might be wrong. Pls have a second thought.