Basic Mathematics & Logarithm
Logarithmic Inequalities
Grade 11
Question:
<p>Which of the following is correct?</p>
<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: Logarithmic inequalities flip their direction when the base is between 0 and 1, and logarithmic identities require careful verification of domain restrictions and base properties.
<p><strong>Step 1: Check logarithmic properties with base > 1</strong></p><p>For base > 1: log_b(x₁) > log_b(x₂) ⟺ x₁ > x₂ (inequality preserves)</p><p><strong>Step 2: Check logarithmic properties with 0 < base < 1</strong></p><p>For 0 < b < 1: log_b(x₁) > log_b(x₂) ⟺ x₁ < x₂ (inequality reverses)</p><p><strong>Step 3: Verify fundamental identities</strong></p><p>• log_b(xy) = log_b(x) + log_b(y) ✓ (for x,y > 0)</p><p>• log_b(x^n) = n·log_b(x) ✓ (for x > 0)</p><p>• log_b(1/x) = -log_b(x) ✓ (for x > 0)</p><p><strong>Step 4: Test specific inequality cases</strong></p><p>If comparing log₀.₅(2) vs log₀.₅(3): Since 0 < 0.5 < 1 and 2 < 3, we have log₀.₅(2) > log₀.₅(3) ✓</p><p>∴ Answer: A,B,C (All standard logarithmic properties and inequality reversals are correct)</p>
Correct Answer: A,B,C