For n = 1, this can be formulated using the approach by @Michal_Adamaszek in Can CVX solve this kind of function {x-log(1+x/(x+1))} , resulting in log(2-inv_pos(1+x1))
If this can be extended to n >= 2, it looks tricky, if possible at all, because each additive log term is not concave, so it seems the whole function will have to be dealt with at once. Also, for high enough values of x (some amount greater than 1), the whole function is not concave, so it seems that will be a complicating matter in the formulation. Also, I believe for n >= 2, the argument of log is not concave, which suggests to me the approach in the 1st paragraph is a dead end for n >= 2.