Convexity of a log quadratic function

Question

My objective function is $\log_2(x^2+x+1)$. Is this a quasiconvex function? If not, is it possible to rewrite it as a convex function? Thanks!

Answer 1

If $f$ is convex then $g=\log_2 f$ is quasiconvex. Indeed $$g(tx +(1-t)y)=\log_2 f(tx +(1-t)y)\leq \log_2 (tf(x) +(1-t)f(y)) \leq \log_2\max\{f(x),f(y)\}=\max\{\log_2 f(x),\log_2 f(y)\}=\max\{g(x),g(y)\}$$