$\text{A.}$ $\exists x \in(0,+\infty), 2^x>3^x$
$\text{B.}$ $\exists x \in(0,1), \log _2 x < \log _3 x$
$\text{C.}$ $\forall x \in(0,+\infty),\left(\frac{1}{2}\right)^x>\log _{\frac{1}{3}} x$
$\text{D.}$ $\forall x \in\left(0, \frac{1}{3}\right),\left(\frac{1}{2}\right)^x < \log _{\frac{1}{3}} x$
$\text{E.}$
$\text{F.}$