Basic Mathematics & Logarithm
Logarithmic Expressions
Grade 11
Question:
<p>Let \(x = (\text{antilog}_2 3) \cdot \log_3 2\), \(y = \log_2(\log_3(\log_2 512))\) and \(z = \log_5 3 \cdot \log_7 5 \cdot \log_2 7\), then \(xyz\) is equal to:</p>
Step-by-Step Solution
Key Concept: Recognize that antilog₂ 3 = 2³, use the property logₐ b · logᵦ a = 1, and apply the chain rule for logarithms: logₐ b · logᵦ c · logᶜ d = logₐ d.
<p><strong>Step 1: Calculate x</strong></p><p>x = (antilog₂ 3) · log₃ 2 = 2³ · log₃ 2 = 8 · log₃ 2</p><p>Using the reciprocal property: log₃ 2 = 1/log₂ 3</p><p>Therefore: x = 8/log₂ 3</p><p><strong>Step 2: Calculate y</strong></p><p>First simplify the inner logarithm: log₂ 512 = log₂ 2⁹ = 9</p><p>Then: log₃(log₂ 512) = log₃ 9 = log₃ 3² = 2</p><p>Therefore: y = log₂(2) = 1</p><p><strong>Step 3: Calculate z using the chain rule</strong></p><p>z = log₅ 3 · log₇ 5 · log₂ 7</p><p>By the chain rule (logₐ b · logᵦ c · logᶜ d = logₐ d):</p><p>z = log₂ 3</p><p><strong>Step 4: Calculate xyz</strong></p><p>xyz = (8/log₂ 3) · 1 · (log₂ 3)</p><p>xyz = 8</p><p>∴ <strong>Answer: 8</strong></p>
Correct Answer: 8