Is it true for concave functions?

Question

I have a function $f(\textbf{x})$ which is a concave function, ($\textbf{x}$ is a two element vector whose entities can take any value from the real line). I think since $\log(x)$ is a concave function therefore $\log(f(\textbf{x}))$ should be a concave function too. Is my reasoning right or wrong? Thanks in advance.

Answer 1

If $f$ is defined on a convex domain, and $f(\mathbf{x}) > 0$ for all $\mathbf{x}$ in that domain, then the answer is yes.

In fact, it is a result of a well-known theorem: for any concave functions $f, g$, provided that the composition $h(\mathbf{x}) = g(f(\mathbf{x}))$ is defined on the domain of $f$, and $g$ is increasing on the range of $f$, the composition $h$ is also concave on the domain of $f$. In your case, $g \equiv \log$, which is indeed concave and increasing.