Basic Mathematics & Logarithm
Logarithmic Inequalities
Grade 11
Question:
<p>If \(\left(\dfrac{1}{x}\right)^{\lambda} \leq \left(\dfrac{1}{9}\right)^{r}\), then which of the following correctly describes the relationship between \(x\), \(\lambda\), and \(r\)?</p>
<p>(a) \(\dfrac{x}{\lambda} \geq \dfrac{\ln x}{\ln 9}\)</p>
<p>(b) \(\dfrac{x}{\lambda} \leq \dfrac{\ln x}{\ln 9}\)</p>
<p>(c) \(x \ln 9 \geq \lambda \ln x\)</p>
<p>(d) \(\dfrac{x}{\lambda} \leq \dfrac{\ln x}{\ln 9}\)</p>
Step-by-Step Solution
Key Concept: When bases are reciprocals (1/x and 1/9), rewrite both sides with the same base to compare exponents directly. For 0 < base < 1, inequality direction reverses when comparing exponents.
<p><strong>Step 1:</strong> Rewrite both sides using a common base approach.</p><p>Given: (1/x)^λ ≤ (1/9)^r</p><p><strong>Step 2:</strong> Rewrite (1/9)^r as 9^(-r), so: (1/x)^λ ≤ 9^(-r)</p><p><strong>Step 3:</strong> Express 9 as 3²: (1/x)^λ ≤ (3²)^(-r) = 3^(-2r)</p><p><strong>Step 4:</strong> For the inequality to hold with reciprocal bases, write (1/x)^λ = x^(-λ).</p><p><strong>Step 5:</strong> If x = 3, then: 3^(-λ) ≤ 3^(-2r)</p><p><strong>Step 6:</strong> Since base 3 > 1, compare exponents directly: -λ ≤ -2r, which gives <strong>λ ≥ 2r</strong></p><p>∴ Answer: D</p>
Correct Answer: D