The value of $\log_{\sqrt{2}} 32$ is

Concept & Formula When the base is a surd (root), convert the logarithmic equation to an exponential form to easily compare the powers of a common integer base. $$ \log_a b = x \implies a^x = b $$ Step-by-Step Solution * **Given:** Evaluate $\log_{\sqrt{2}} 32$. Let this expression equal $x$. $$ \log_{\sqrt{2}} 32 = x $$ * **Calculation:** Rewrite the expression in exponential form: $$ (\sqrt{2})^x = 32 $$ * Express both sides using the common base of $2$. We know $\sqrt{2} = 2^{1/2}$ and $32 = 2^5$: $$ (2^{1/2})^x = 2^5 $$ * Apply the power of a power rule $(a^m)^n = a^{m \cdot n}$: $$ 2^{x/2} = 2^5 $$ * Since the bases are identical, equate the exponents: $$ \frac{x}{2} = 5 $$ $$ x = 10 $$ Exam Strategy & Shortcut Think of it this way: It takes $5$ twos to make $32$ ($2^5 = 32$). Since the base is $\sqrt{2}$, you need exactly twice as many to reach the same value. $5 \times 2 = 10$. Common Pitfall Students often write the answer as $5/2$, incorrectly dividing the power of $32$ ($5$) by $2$ instead of multiplying. Remember that a smaller base requires a larger exponent. Final Answer **Therefore, the correct answer is 10.**