<p>If \(x_1\) and \(x_2\) are the solutions of the equation \(5^{(\log_5 x)^2} + x^{\log_5 x} = 1250\), then \(x_1 \cdot x_2\) is equal to:</p>
Step-by-Step Solution
Key Concept: Substitute y = log₅(x) to convert the exponential equation into a polynomial form, recognizing that 5^((log₅ x)²) = x^(log₅ x) when exponents are equal.
<p><strong>Step 1:</strong> Let y = log₅(x), so x = 5^y</p><p><strong>Step 2:</strong> Rewrite the original equation using this substitution:</p><p>• 5^((log₅ x)²) = 5^(y²)</p><p>• x^(log₅ x) = (5^y)^y = 5^(y²)</p><p>So the equation becomes: 5^(y²) + 5^(y²) = 1250</p><p><strong>Step 3:</strong> Simplify: 2·5^(y²) = 1250</p><p>Therefore: 5^(y²) = 625 = 5⁴</p><p><strong>Step 4:</strong> This gives y² = 4, so y = 2 or y = -2</p><p><strong>Step 5:</strong> Convert back to x:</p><p>• When y = 2: x₁ = 5² = 25</p><p>• When y = -2: x₂ = 5^(-2) = 1/25</p><p><strong>Step 6:</strong> Verification (for y = 2): 5⁴ + 25² = 625 + 625 = 1250 ✓</p><p><strong>Step 7:</strong> Calculate x₁·x₂ = 25 × (1/25) = 1</p><p>∴ Answer: <strong>1</strong></p>
Correct Answer: 1