Is $\text{Log}(f(x))$ concave in $x$ if $f(x)$ is both positive and quasi-concave in $x$?

Question

Is it true that $\text{Log}$ of some postive and quasi-concave function is always a concave function?

i-e If $f:x\mapsto \mathbb{R^+}$ where $f$ is quasi-concave in $x \implies$ $\text{Log}(f(x))$ is concave in $x$? considering that $f \in \mathcal C^2.$

For instance normal distribution, $\mathcal{N}(x|\mu,\sigma)$, is both quasi-concave and log-concave in corresponding convex domain, and therefore $\text{Log}(\mathcal{N}(x|\mu,\sigma))$ is concave in $x$. Is it true in general too?

Answer 1

I do not believe this is true in general.

Consider the function

$$f(x) = 1-\frac{\sqrt{|x|}}{4\sqrt{|x|}+1}.$$

This is quasi-concave (but not concave). Now look at the graph of $log(f(x))$.

http://www.wolframalpha.com/input/?i=log(1-sqrt(%7Cx%7C)%2F(4sqrt(%7Cx%7C)%2B1)))

Visually, one can see that it is quasi concave but not concave.