<p>If \(1,\ \log_3\sqrt{3^{1-x}+2},\ \log_3(4\cdot 3^x - 1)\) are in AP, then \(x\) equals</p>
Step-by-Step Solution
Key Concept: If three terms are in AP, then the middle term equals the average of the first and third terms: 2b = a + c. Use this property along with logarithm properties to create an equation in terms of 3^x.
<p><strong>Step 1:</strong> Apply the AP condition. If a, b, c are in AP, then 2b = a + c.</p><p>Let a = 1, b = log₃√(3^(1-x)+2), c = log₃(4·3^x - 1)</p><p>2·log₃√(3^(1-x)+2) = 1 + log₃(4·3^x - 1)</p><p><strong>Step 2:</strong> Simplify the left side using logarithm properties.</p><p>2·(1/2)log₃(3^(1-x)+2) = 1 + log₃(4·3^x - 1)</p><p>log₃(3^(1-x)+2) = 1 + log₃(4·3^x - 1)</p><p><strong>Step 3:</strong> Convert to exponential form and simplify.</p><p>log₃(3^(1-x)+2) = log₃3 + log₃(4·3^x - 1)</p><p>log₃(3^(1-x)+2) = log₃[3(4·3^x - 1)]</p><p>3^(1-x) + 2 = 12·3^x - 3</p><p><strong>Step 4:</strong> Substitute y = 3^x and solve.</p><p>3·3^(-x) + 2 = 12·3^x - 3</p><p>3/y + 2 = 12y - 3</p><p>3 + 2y = 12y² - 3y</p><p>12y² - 5y - 3 = 0</p><p>(3y + 1)(4y - 3) = 0</p><p><strong>Step 5:</strong> Find valid solution.</p><p>y = -1/3 (rejected, since 3^x > 0) or y = 3/4</p><p>3^x = 3/4 → x = log₃(3/4) = log₃3 - log₃4 = 1 - log₃4</p><p>∴ Answer: B</p>
Correct Answer: B