Basic Mathematics & Logarithm
Logarithm Properties
Grade 11
Question:
<p>If <strong>f</strong>(<strong>x</strong>) = <strong>ax</strong><sup>2</sup> − <strong>bx</strong> + <strong>c</strong> has two distinct roots α and β, and f(0) and f(1) are of the same sign, with constraints leading to a ≥ 5, b ≥ 5, c ≥ 1, find the least value of log<sub>b</sub>(abc).</p>
Step-by-Step Solution
Key Concept: Find the minimum product abc using the bounds on individual coefficients, then apply logarithm properties to find the minimum logarithmic value.
<p><strong>Step 1:</strong> From previous analysis: a ≥ 5, b ≥ 5, c ≥ 1</p><p><strong>Step 2:</strong> Therefore abc ≥ 5 × 5 × 1 = 25</p><p><strong>Step 3:</strong> Taking logarithm base b: log<sub>b</sub>(abc) ≥ log<sub>b</sub>(25) = log<sub>5</sub>(25) = 2 (when b = 5)</p><p><strong>Step 4:</strong> The minimum is achieved when a = 5, b = 5, c = 1, giving abc = 25</p><p>∴ <strong>Least value of log<sub>b</sub>(abc) is 2</strong></p>
Correct Answer: 2