Concavity of a simple objective function

Question

I have a maximization problem with the following objective function $$ f= \log\left(\sum_{i=1}^nx_i\right)+\log\left(1-\sum_{i=1}^n\frac{x_i}{y_i}\right). $$ I would like to show that $f$ is concave in general. It is when $n=1$ since then $$ f=\log\left(x-\frac{x^2}{y} \right) $$ and $x^2/y$ is convex. Can we generalize this result to $n\geq 2$?