Basic Mathematics & Logarithm
Nested Radicals and Logarithmic Equations
Grade 11
Question:
<p>If <span class='math'>a = \sqrt{3 + 2\sqrt{2}}</span>, <span class='math'>b = \sqrt{3 - 2\sqrt{2}}</span>, then the value of <span class='math'>\log_a b</span> is equal to</p>
<p>(P) -1</p>
<p>(Q) 1</p>
<p>(R) 2</p>
<p>(S) 3</p>
<p>(T) None of these</p>
Step-by-Step Solution
Key Concept: Recognize that the denested values are reciprocals, so the logarithm equals -1.
<p><strong>Step 1:</strong> Simplify <span class='math'>a = \sqrt{3 + 2\sqrt{2}}</span>.</p><p>Note that <span class='math'>3 + 2\sqrt{2} = (\sqrt{2} + 1)^2</span>.</p><p>So <span class='math'>a = \sqrt{2} + 1</span>.</p><p><strong>Step 2:</strong> Simplify <span class='math'>b = \sqrt{3 - 2\sqrt{2}}</span>.</p><p>Note that <span class='math'>3 - 2\sqrt{2} = (\sqrt{2} - 1)^2</span>.</p><p>So <span class='math'>b = \sqrt{2} - 1</span>.</p><p><strong>Step 3:</strong> Calculate <span class='math'>\log_a b = \log_{\sqrt{2}+1}(\sqrt{2}-1)</span>.</p><p>Note that <span class='math'>(\sqrt{2} + 1)(\sqrt{2} - 1) = 2 - 1 = 1</span>, so <span class='math'>\sqrt{2} - 1 = \frac{1}{\sqrt{2}+1} = (\sqrt{2}+1)^{-1}</span>.</p><p>Therefore <span class='math'>\log_a b = \log_a(a^{-1}) = -1</span>.</p><p>∴ Answer is P (-1).</p>
Correct Answer: P