4. Let points $P, Q$ be on the graphs of the functions $y=2^{x}$ and $y=\log _{2} x$, respectively, then the minimum value of $|P Q|=$
荟事 $\frac{1+\ln (\ln 2)}{\ln 2} \sqrt{2}$.

Gather matters $\frac{1+\ln (\ln 2)}{\ln 2} \sqrt{2}$.