Calculating complexity of logarithm

Question

As known the complexity to insert a node into AVL tree is log(c) when c is the number of nodes inside the tree. I'm looking to insert m nodes into the tree so the complexity is:

log(c)+log(c+1)+...+log(c+m)

Any ideas/suggestions on how I can solve this and get Big-O?

Answer 1

Assuming c is not a constant:

log(c)+log(c+1)+...+log(c+m) = log(c * (c+1) * .... * (c+m))
                     = log((m+c)!/c!)
                     = log((m+c)!) - log(c!)
                     = O_theta((m+c)log(m+c)- clogc)

But for the sake of the Big-O complexities, we usually quote the loose-bounds: O(NlogN) where N=m+c