<p>Which of the following statements are correct?</p><p>(a) \(\log_5\left(\sqrt{7\sqrt{7\sqrt{7\cdots}}}\right) = \log_5 7 > 1\)</p><p>(b) \(\dfrac{1}{\sqrt{3}+\sqrt{2}} < \dfrac{1}{\sqrt{7}+\sqrt{6}} \Rightarrow \sqrt{3}-\sqrt{2} > \sqrt{7}-\sqrt{6}\)</p><p>(c) \(\log_3 10 > 2\) and \(\log_{10} 70 < 2\)</p><p>(d) \(\log_3(3+\sqrt{2}) > 1\) and \(\log_2(2-\sqrt{2}) < 1\)</p>
<p>(a) \(\log_5\left(\sqrt{7\sqrt{7\sqrt{7\cdots}}}\right) = \log_5 7 > 1\)</p>
<p>(b) \(\dfrac{1}{\sqrt{3}+\sqrt{2}} < \dfrac{1}{\sqrt{7}+\sqrt{6}} \Rightarrow \sqrt{3}-\sqrt{2} > \sqrt{7}-\sqrt{6}\)</p>
<p>(c) \(\log_3 10 > 2\) and \(\log_{10} 70 < 2\)</p>
<p>(d) \(\log_3(3+\sqrt{2}) > 1\) and \(\log_2(2-\sqrt{2}) < 1\)</p>
Step-by-Step Solution
Key Concept: For nested radicals, recognize the pattern: √(7√(7√(7...))) converges to 7, making log₅(7) < 1 since 7 < 25. For inequalities, use base properties: logₐ(b) > 1 iff b > a (when a > 1), and compare logarithms by converting to common bases or using monotonicity.
<p><strong>Statement (a): log₅(√(7√(7√(7...)))) = log₅(7) > 1</strong></p><p>Let x = √(7√(7√(7...))). Then x = √(7x), so x² = 7x, giving x = 7 (taking positive root).</p><p>log₅(7) compared to 1: Since 5¹ = 5 and 7 > 5, we have log₅(7) > 1. ✓ <strong>CORRECT</strong></p><p><strong>Statement (b): 1/(√3+√2) vs √7-√6</strong></p><p>Rationalize: 1/(√3+√2) = (√3-√2)/((√3+√2)(√3-√2)) = (√3-√2)/(3-2) = √3-√2.</p><p>Compare √3-√2 with √7-√6: Cross-multiply (√3+√6)² = 9+6√18 vs (√2+√7)² = 9+4√14.</p><p>Since 6√18 = 18√2 ≈ 25.46 and 4√14 ≈ 14.97, we have √3-√2 < √7-√6. ✓ <strong>CORRECT</strong></p><p><strong>Statement (c): log₃(10) > 2 and log₁₀(70) < 2</strong></p><p>log₃(10) > 2? Check: 3² = 9 < 10, so log₃(10) > 2. ✓</p><p>log₁₀(70) < 2? Check: 10² = 100 > 70, so log₁₀(70) < 2. ✓ <strong>CORRECT</strong></p><p><strong>Statement (d): log₃(3+√2) > 1 and log₂(2-√2) < 0</strong></p><p>log₃(3+√2) > 1? Check: 3¹ = 3 and 3+√2 > 3, so log₃(3+√2) > 1. ✓</p><p>log₂(2-√2) < 0? Check: 2⁰ = 1 and 0 < 2-√2 ≈ 0.586 < 1, so log₂(2-√2) < 0. ✓ <strong>CORRECT</strong></p><p>∴ Answer: <strong>A, B, C, D</strong></p>
Correct Answer: A, B, C, D