Answer
$x=-\frac{1}{2}$
Work Step by Step
When the properties of exponents are used, $\dfrac{1}{\sqrt2} = 2^{-\frac{1}{2}}$.
Thus, the given expression is equivalent to
$\log_2{(2^{-\frac{1}{2}})}$
Let
$\log_{2}{(2^{-\frac{1}{2}})} = x$
RECALL:
$\log_b{y} =x \longrightarrow b^x=y$
Use the rule above, where $y=2^{-\frac{1}{2}}$ and $b=2$, to obtain:
$2^x=2^{-\frac{1}{2}}$
Use the rule "If $a^x=a^y$, then $x=y$" to obtain:
$x=-\frac{1}{2}$