Answer
Please see the work below.
Work Step by Step
We know that
$\tau =0$
$\implies (\frac{1}{2}mg)a-(mg)b=0$
This simplifies to:
$a=2b$
$\implies a+b=3b$
but $3b=\frac{L}{2}$
$\implies b=\frac{L}{6}$
Now $x=a+b$
We plug in the known values to obtain:
$x=\frac{L}{2}+\frac{L}{6}$
$\implies x=\frac{2}{3}L$