Basic Mathematics & Logarithm
Nested Radicals and Logarithmic Equations
Grade 11
Question:
<p>If <span class='math'>a = 3(\sqrt{8 + 2\sqrt{7}} - \sqrt{8 - 2\sqrt{7}})</span>, <span class='math'>b = \sqrt{(42)(30) + 36}</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: Denest the nested radicals by recognizing perfect square forms, then evaluate the logarithm.
<p><strong>Step 1:</strong> Simplify <span class='math'>a = 3(\sqrt{8 + 2\sqrt{7}} - \sqrt{8 - 2\sqrt{7}})</span>.</p><p>Note that <span class='math'>8 + 2\sqrt{7} = (\sqrt{7} + 1)^2</span> and <span class='math'>8 - 2\sqrt{7} = (\sqrt{7} - 1)^2</span>.</p><p>So <span class='math'>a = 3(\sqrt{7} + 1 - (\sqrt{7} - 1)) = 3(2) = 6</span>.</p><p><strong>Step 2:</strong> Simplify <span class='math'>b = \sqrt{(42)(30) + 36} = \sqrt{1260 + 36} = \sqrt{1296} = 36</span>.</p><p><strong>Step 3:</strong> Calculate <span class='math'>\log_a b = \log_6 36 = \log_6 6^2 = 2</span>.</p><p>∴ Answer is R (2).</p>
Correct Answer: R