Basic Mathematics & Logarithm
Inequalities
Grade 11
Question:
<p>Which of the following is always true?</p>
<p>(a) If \(a < b\), then \(a^2 < b^2\)</p>
<p>(b) If \(a < b\), then \(\dfrac{1}{a} > \dfrac{1}{b}\)</p>
<p>(c) If \(a < b\), then \(|a| < |b|\)</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Logarithm properties depend critically on the base value. For base b > 1, log_b is an increasing function, while for 0 < b < 1, it's decreasing. Without knowing the base, we cannot make universal claims about logarithmic inequalities.
<p><strong>Step 1:</strong> Recognize that logarithmic behavior depends fundamentally on whether the base b > 1 or 0 < b < 1.</p><p><strong>Step 2:</strong> For b > 1: log_b is increasing, so x₁ < x₂ ⟹ log_b(x₁) < log_b(x₂)</p><p><strong>Step 3:</strong> For 0 < b < 1: log_b is decreasing, so x₁ < x₂ ⟹ log_b(x₁) > log_b(x₂) (inequality reverses)</p><p><strong>Step 4:</strong> Check universal identities that hold for ALL valid bases: log_b(1) = 0, log_b(b) = 1, and log_b(xy) = log_b(x) + log_b(y) are always true regardless of base.</p><p><strong>Step 5:</strong> Any statement comparing logarithmic values without specifying base cannot be 'always true' unless it's a fundamental logarithmic identity.</p><p>∴ Answer: D (The correct option must be a base-independent logarithmic identity)</p>
Correct Answer: D