If $\log_2 [\log_3 (\log_2 x)] = 1$, then $x$ is equal to :

Concept & Formula This problem requires "unwinding" nested logarithms by repeatedly converting them from logarithmic form to exponential form. The foundational principle applied at each layer is: $$\log_a (B) = C \implies B = a^C$$ By doing this iteratively, we isolate the variable $x$. Step-by-Step Solution * **Given:** $\log_2 [\log_3 (\log_2 x)] = 1$ * **Unwind the First (Outer) Layer:** Convert the outermost base-2 logarithm into an exponential expression. $$\log_3 (\log_2 x) = 2^1$$ $$\log_3 (\log_2 x) = 2$$ * **Unwind the Second Layer:** Now convert the base-3 logarithm into an exponential expression. $$\log_2 x = 3^2$$ $$\log_2 x = 9$$ * **Unwind the Third (Inner) Layer:** Finally, convert the innermost base-2 logarithm to solve for $x$. $$x = 2^9$$ * **Calculate Final Value:** Evaluate the power of 2. $$x = 512$$ Exam Strategy & Shortcut Think of this as peeling an onion. Read the equation from left to right and push each base to the opposite side as a base for an exponent. Mental flow: Base 2 swings over to make $2^1=2$. Base 3 swings over to make $3^2=9$. Base 2 swings over to make $2^9=512$. This chain of exponents can be done mentally without writing intermediate equations. Common Pitfall A common mistake is arithmetic error when evaluating the final power. Students might accidentally compute $3^2$ as 6 instead of 9, leading to $x = 2^6 = 64$ (which isn't an option here, but is a common distractor). Always double-check basic exponential calculations. Final Answer **Therefore, the correct answer is 512.**