<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: if log₂(A) = 1, then A = 2. Then solve log₃(x²) = 2 to get x² = 9, yielding two distinct real solutions.
<p><strong>Step 1:</strong> Apply the definition of logarithm to the outer equation.</p><p>If log₂(log₃(x²)) = 1, then log₃(x²) = 2¹ = 2</p><p><strong>Step 2:</strong> Apply the definition of logarithm again to solve for x².</p><p>If log₃(x²) = 2, then x² = 3² = 9</p><p><strong>Step 3:</strong> Solve the quadratic equation x² = 9.</p><p>x = 3 or x = -3</p><p><strong>Step 4:</strong> Verify domain restrictions.</p><p>For log₃(x²) to be defined: x² > 0 ✓ (true for both values)</p><p>For log₂(log₃(x²)) to be defined: log₃(x²) > 0, which means x² > 1, so |x| > 1 ✓ (both |3| and |-3| equal 3 > 1)</p><p>Verification: log₃(9) = 2, and log₂(2) = 1 ✓</p><p>∴ Answer: <strong>2</strong> (Both x = 3 and x = -3 satisfy the equation)</p>
Correct Answer: C