<p><strong>70.</strong> Find the value of the expression \(\frac{2}{\log_4(2000)} + \frac{3}{\log_5(2000)}\).</p>
Step-by-Step Solution
Key Concept: Use the change of base formula to convert reciprocals of logarithms into logarithms with a common base. Specifically, 1/log_a(b) = log_b(a), which allows us to rewrite the expression in terms of logarithms base 2000.
<p><strong>Step 1:</strong> Apply the change of base formula property: 1/log_a(b) = log_b(a).</p><p>We know that if log_a(b) = x, then 1/x = log_b(a).</p><p>Therefore: 1/log_4(2000) = log_2000(4) and 1/log_5(2000) = log_2000(5)</p><p><strong>Step 2:</strong> Rewrite the original expression.</p><p>2/log_4(2000) + 3/log_5(2000) = 2·log_2000(4) + 3·log_2000(5)</p><p><strong>Step 3:</strong> Use the logarithm power rule: k·log_a(b) = log_a(b^k).</p><p>= log_2000(4²) + log_2000(5³)</p><p>= log_2000(16) + log_2000(125)</p><p><strong>Step 4:</strong> Apply the logarithm product rule: log_a(x) + log_a(y) = log_a(xy).</p><p>= log_2000(16 × 125)</p><p>= log_2000(2000)</p><p><strong>Step 5:</strong> Evaluate log_2000(2000).</p><p>By definition, log_a(a) = 1 for any valid base a.</p><p>Therefore: log_2000(2000) = 1</p><p><strong>∴ Answer: 1</strong></p>
Correct Answer: 1