Basic Mathematics & Logarithm
Logarithmic simplification
Grade 11
Question:
<p>The value of \( \log_6\left(\sqrt{2-\sqrt{3}} + \sqrt{2+\sqrt{3}}\right) \) is:</p>
<p>A) 1</p>
<p>B) \(\frac{1}{2}\)</p>
<p>C) 2</p>
<p>D) \(\frac{1}{3}\)</p>
Step-by-Step Solution
Key Concept: Recognize that the expression under the logarithm can be simplified by squaring it first, then using the identity (a+b)² = a² + b² + 2ab to find the sum directly without computing individual square roots.
<p><strong>Step 1:</strong> Let S = √(2-√3) + √(2+√3). Square both sides to simplify.</p><p><strong>Step 2:</strong> S² = (2-√3) + (2+√3) + 2√((2-√3)(2+√3))</p><p><strong>Step 3:</strong> S² = 4 + 2√(4-3) = 4 + 2(1) = 6</p><p><strong>Step 4:</strong> Since S > 0, we have S = √6</p><p><strong>Step 5:</strong> log₆(√6) = log₆(6^(1/2)) = (1/2)log₆(6) = 1/2</p><p>∴ Answer: <strong>1/2</strong></p>
Correct Answer: B