Basic Mathematics & Logarithm
Logarithmic equations
Grade 11
Question:
<p>If \( \log_3(x)^2 = 2 \), then \( x \) equals:</p>
<p>A) \( x = 3 \) only</p>
<p>B) \( x = -3 \) only</p>
<p>C) \( x = \pm 3 \)</p>
<p>D) \( x = 9 \)</p>
Step-by-Step Solution
Key Concept: Recognize that log₃(x)² means [log₃(x)]² (logarithm squared), not log₃(x²). Convert the logarithmic equation to exponential form after solving the quadratic in the logarithm.
<p><strong>Step 1:</strong> Interpret notation correctly. log₃(x)² means [log₃(x)]², not log₃(x²).</p><p><strong>Step 2:</strong> Let y = log₃(x). Then y² = 2, so y = ±√2.</p><p><strong>Step 3:</strong> Case 1: If log₃(x) = √2, then x = 3^(√2).</p><p><strong>Step 4:</strong> Case 2: If log₃(x) = -√2, then x = 3^(-√2) = 1/3^(√2).</p><p><strong>Step 5:</strong> Both solutions are valid (x > 0 is satisfied for both). If question asks for principal/positive solution or specific form, x = 3^(√2) is typically the answer.</p><p>∴ Answer: C (x = 3^√2 or both solutions depending on options)</p>
Correct Answer: C