<p>If <br>\(\frac{1}{3}\log_3 a + \frac{1}{2}\log_3 b = \frac{7}{2}\)<br>\(\frac{1}{2}\log_3 a + \frac{1}{3}\log_3 b = \frac{2}{3}\)<br>then find the value of \(ab\).</p>
Step-by-Step Solution
Key Concept: Convert the logarithmic equations into a system of linear equations by substituting x = log₃ a and y = log₃ b, then solve for x + y to find log₃(ab). Use the property that log₃(ab) = log₃ a + log₃ b.
<p><strong>Step 1:</strong> Let x = log₃ a and y = log₃ b. The equations become:</p><p>⅓x + ½y = 7/2 ... (1)</p><p>½x + ⅓y = 2/3 ... (2)</p><p><strong>Step 2:</strong> Add equations (1) and (2):</p><p>(⅓ + ½)x + (½ + ⅓)y = 7/2 + 2/3</p><p>5/6·x + 5/6·y = 21/6 + 4/6 = 25/6</p><p>5/6(x + y) = 25/6</p><p>x + y = 5</p><p><strong>Step 3:</strong> Since x + y = log₃ a + log₃ b = log₃(ab) = 5</p><p>Therefore: ab = 3⁵ = 243</p>
Correct Answer: 243