Answer
$\log_\sqrt{2}{4} = 4$
Work Step by Step
Let
$\log_\sqrt{2}{4}=y$
RECALL:
$\log_a{x} = y \longrightarrow a^y=x$
Use this rule to obtain:
$\log_\sqrt2{4} = y \longrightarrow (\sqrt{2})^y=4$
Write $4$ as $2^2$ to obtain:
$(\sqrt{2})^y=2^2$
Note that $\sqrt{2} = 2^{\frac{1}{2}}$. Thus, the equation above is equivalent to:
$(2^{\frac{1}{2}})^y = 2^2$
Use the rule $(a^m)^n=a^{mn}$ to obtain:
$2^{\frac{1}{2} \cdot y} = 2^2
\\2^{\frac{y}{2}} = 2^2$
Use the rule "$a^m=a^n \longrightarrow m=n$" to obtain:
$\frac{y}{2} = 2$
Multiply $2$ to both sides of the equation to obtain:
$2(\frac{y}{2}) = 2(2)
\\y = 4$
Thus,
$\log_\sqrt{2}{4} = 4$