Prove that if a graph G VE is connected then log E BigOlog

Prove that if a graph G = (V,E) is connected, then log |E| = BigO(log |V|).

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If the graph is connected then |E| = |V|*(|V|-1)/2

log |E| = log(|V|*(|V|-1)/2) = log(|V|) + log(|V|-1) - log(2)

We know that for |V|>2:

log(|V|) > log(|V|-1)

So adding log(|V|) on both sides,

2*log(|V|) > log(|V|) + log(|V|-1)

As log(2) > 0 So

2*log(|V|) > log(|V|) + log(|V|-1) - log(2)

So 2*log(|V|) > log |E|

So for c = 2 and |V|>2

log|E| = O(log|V|)