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If x₁ and x₂ satisfy the equation x^log ₁₀ ^x=100x, then the value of x₁x₂is???

$x^{\log_{10}} x = 100 x \Rightarrow \log_{10} x = \log_{x} (100 x) \Rightarrow \log_{10} x = \log_{x} 100 + 1 \Rightarrow \log_{10} x = 2 \log_{x} 10 + 1 \Rightarrow \log_{10} x = \frac{2}{\log_{10} x} + 1 \Rightarrow (\log_{10} x)^{2} = 2 + \log_{10} x \Rightarrow (\log_{10} x)^{2} - \log_{10} x - 2 = 0 \Rightarrow (\log_{10} x - 2) (\log_{10} x + 1) = 0 \Rightarrow \log_{10} x = 2 or \log_{10} x = - 1 \Rightarrow x = 100 or x = \frac{1}{10} \Rightarrow x_{1} = 100 and x_{2} = \frac{1}{10} ∴ x_{1} x_{2} = 10$

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