<p>If \(\log_2(5 \times 2^x + 1)\), \(\log_4(2^{1-x} + 1)\) and 1 are in A.P., then <i>x</i> equals</p>
Step-by-Step Solution
Key Concept: Convert all logarithms to the same base and use the A.P. condition: middle term = (first term + third term)/2. This transforms the logarithmic equation into an algebraic equation in terms of 2^x.
<p><strong>Step 1:</strong> Convert to common base. Let log₂(5×2^x + 1) = a, log₄(2^(1-x) + 1) = b, and 1 = c.</p><p><strong>Step 2:</strong> Convert b to base 2: log₄(2^(1-x) + 1) = log₂(2^(1-x) + 1)/log₂(4) = log₂(2^(1-x) + 1)/2</p><p><strong>Step 3:</strong> Apply A.P. condition: 2b = a + c</p><p>2 × [log₂(2^(1-x) + 1)/2] = log₂(5×2^x + 1) + 1</p><p>log₂(2^(1-x) + 1) = log₂(5×2^x + 1) + log₂(2)</p><p><strong>Step 4:</strong> Simplify using logarithm properties:</p><p>log₂(2^(1-x) + 1) = log₂(2(5×2^x + 1))</p><p>2^(1-x) + 1 = 10×2^x + 2</p><p><strong>Step 5:</strong> Let y = 2^x. Then 2^(1-x) = 2/y:</p><p>2/y + 1 = 10y + 2</p><p>2/y = 10y + 1</p><p>2 = 10y² + y</p><p>10y² + y - 2 = 0</p><p><strong>Step 6:</strong> Factor: (5y + 2)(2y - 1) = 0</p><p>y = -2/5 (rejected, since 2^x > 0) or y = 1/2</p><p>2^x = 1/2 = 2^(-1)</p><p>∴ x = -1 (Answer: D)</p>
Correct Answer: D