Given: angle between $\overrightarrow a$ and $\overrightarrow b$ is acute.
$\Rightarrow\:\overrightarrow a.\overrightarrow b>0$
$\Rightarrow\:c(log_3x)^2-12+6c log_3x>0$ $\forall x\in (0,\infty)$
Let $y=log_3x$
$\Rightarrow\: cy^2+6cy-12>0$
A quadratic expression $ax^2+bx+c$ is positive if $a>0$ and $b^2-4ac<0$
$\therefore\:c>0 $ and $ 36c^2+48c<0$
But if $c>0,$ then $36c^2+48c$ cannot be $<0$
$\Rightarrow $ The set of values of $c$ is $\phi$, the null set.