How can I convert x*log(u(x)) into a concave function, given u(x) is concave w.r.t. x???

That is not necessarily concave. For example, let u(x) = log(x). Then the expression is concave for x <= exp(1), and convex for x >= exp(1).

However, it is concave for some functions u(x). You would have to formulate to work for that specific u(x).

Thank you so much for your kind reply.