<p>Number of value(s) of \(x\) satisfying the equation \(\log_2(\log_3(x^2)) = 1\) is/are:</p>
Step-by-Step Solution
Key Concept: Work backwards from the logarithmic equation by converting from log form to exponential form sequentially, then solve the resulting polynomial equation while respecting domain restrictions at each step.
<p><strong>Step 1:</strong> Convert log₂(log₃(x²)) = 1 to exponential form.</p><p>log₃(x²) = 2¹ = 2</p><p><strong>Step 2:</strong> Convert log₃(x²) = 2 to exponential form.</p><p>x² = 3² = 9</p><p><strong>Step 3:</strong> Solve for x.</p><p>x = ±3</p><p><strong>Step 4:</strong> Verify both solutions satisfy domain restrictions.</p><p>For x = 3: log₃(9) = 2 > 0 ✓, then log₂(2) = 1 ✓</p><p>For x = -3: log₃(9) = 2 > 0 ✓, then log₂(2) = 1 ✓</p><p>Both values are valid since x² = 9 in both cases.</p><p><strong>∴ Answer: 2 values (x = 3 and x = -3)</strong></p>
Correct Answer: C