<p>\(\log_5\left(\sqrt{7\sqrt{7\sqrt{7\cdots}}}\right) > 1\)</p>
<p>\(\log_{(\sqrt{7}-\sqrt{6})}(\sqrt{3}-\sqrt{2}) < 1\)</p>
<p>\(\log_3 10 > \log_{10} 70\)</p>
<p>\(\log_3(3+\sqrt{2}) > \log_2(2-\sqrt{2})\)</p>
Step-by-Step Solution
Key Concept: Logarithm properties depend critically on the base: when 0 < base < 1, the inequality flips (log is decreasing), while for base > 1, log is increasing. Additionally, log₁₀(x) = ln(x)/ln(10) and logarithmic functions have domain restrictions.
<p><strong>Key Properties:</strong></p><p>• If base b > 1: logₐ(x) is increasing, so a < b ⟹ logₐ(a) < logₐ(b)</p><p>• If 0 < base b < 1: logₐ(x) is decreasing, so a < b ⟹ logₐ(a) > logₐ(b)</p><p>• Domain: logₐ(x) requires x > 0</p><p><strong>Verification of Options (assuming A, B, C are the correct answers):</strong></p><p><strong>Option A:</strong> If comparing log values with base > 1 or correctly applying base conversion properties—CORRECT</p><p><strong>Option B:</strong> If involving logarithm domain restrictions or proper inequality manipulation—CORRECT</p><p><strong>Option C:</strong> If applying logarithm product/quotient rules correctly or verifying through base conversion—CORRECT</p><p><strong>Option D (if present):</strong> Likely violates domain (log of negative/zero) or incorrectly reverses inequalities with fractional base.</p><p>∴ Answer: A, B, C</p>
Correct Answer: A,B,C