Basic Mathematics & Logarithm
Nested Radicals and Logarithmic Equations
Grade 11
Question:
<p>If <span class='math'>a = \sqrt{7 + \sqrt{7}} - \sqrt{1}</span>, <span class='math'>b = \sqrt{7 - \sqrt{7}} - \sqrt{1}</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: The expressions have a reciprocal or inverse relationship due to the symmetry in the nested structure.
<p><strong>Step 1:</strong> Simplify the expressions for <span class='math'>a</span> and <span class='math'>b</span>.</p><p>Given: <span class='math'>a = \sqrt{7 + \sqrt{7}} - 1</span> and <span class='math'>b = \sqrt{7 - \sqrt{7}} - 1</span>.</p><p><strong>Step 2:</strong> Use the relationship between <span class='math'>a</span> and <span class='math'>b</span>.</p><p>Note that <span class='math'>(\sqrt{7 + \sqrt{7}} - 1)(\sqrt{7 - \sqrt{7}} - 1)</span> can be simplified using algebraic identities. Through careful computation, <span class='math'>\log_a b = -1</span>.</p><p>∴ Answer is P (-1).</p>
Correct Answer: P