The expression is $\log_{\sqrt{2}}(32)$. Let this value be $y$. By the definition of logarithms, $ \log_b(a) = y$ is equivalent to $b^y = a$. So, we have $(\sqrt{2})^y = 32$. To solve for $y$, we should express both sides of the equation with the same base. The most convenient base here is 2. We know that $\sqrt{2} = 2^{1/2}$. We also know that $32 = 2 \times 2 \times 2 \times 2 \times 2 = 2^5$. Substitute these into the equation: $(2^{1/2})^y = 2^5$. Using the exponent rule $(a^m)^n = a^{mn} $, we get $2^{(1/2) \times y} = 2^5$, which is $2^{y/2} = 2^5$. Since the bases are equal, the exponents must be equal: $\frac{y}{2} = 5$. Multiply both sides by 2: $y = 5 \times 2 = 10$. Thus, $ \log_{\sqrt{2}}(32) = 10$.