The charge on the capacitor at time $t$ in a discharging circuit is given by $q=q_0{e}^{-t/\tau }$

where, $\tau =RC=$ time constant
$\therefore$               $q=q_0{e}^{-t/RC}$

when, $q=\frac{q_0}{2}$

then  $\frac{q_0}{2}=q_0{e}^{-t/RC}$

$\frac{1}{2}=e^{-t/RC}$

Taking natural log on both side ${\log }_e\left(\frac{1}{2}\right)={\log }_0{e}^{-t/RC}$

$-{\log }_{e}2=-\frac{t}{RC}{\log }_{e}e$

$\Rightarrow t=RC{\log }_{e}2$