Chapter 4 - Exponential and Logarithmic Functions - Section 4.4 Logarithmic Functions - 4.4 Assess Your Understanding - Page 321: 33

$4$

Note that $y=\log_a x \text{ is equivalent to } x= a^y$. Thus, if $y = \log_{\sqrt{2}} 4 \hspace{5pt},$ then $\hspace{5pt}(\sqrt{2})^y=4$ Since $4=2^2=(\sqrt{2})^4$, then $(\sqrt{2})^y=(\sqrt{2})^4$ Use the rule $a^m=a^n \implies m=n$ to obtain: $y=4$ Therefore, $ \log_{\sqrt{2}} 4 = \boxed{4}$