$Product of roots of the equation \log_{5} (5^{1 / x} + 125) = \log_{5} 6 + 1 + \frac{1}{2 x} is$

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$\frac{1}{6}$

$\frac{1}{8}$

$\frac{1}{2}$

$\frac{1}{4}$

answer is C.

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Detailed Solution

$\begin{array}{l} \log_{5} (5^{1 / x} + 125) = \log_{5} 6 + 1 + \frac{1}{2 x} \\ \Rightarrow \log_{5} (5^{1 / x} + 125) - \log_{5} 6 = 1 + \frac{1}{2 x} \\ \Rightarrow \log_{5} (\frac{5^{\frac{1}{x}} + 125}{6}) = 1 + \frac{1}{2 x} \\ \Rightarrow \frac{5^{\frac{1}{x}} + 125}{6} = 5^{1 + \frac{1}{2 x}} \\ \Rightarrow \frac{5^{\frac{1}{x}} + 125}{6} = 5 \times 5^{\frac{1}{2 x}} \\ \Rightarrow \frac{y^{2} + 125}{6} = 5 y (where y = 5^{\frac{1}{2 x}}) \\ \Rightarrow y^{2} - 30 y + 125 = 0 \\ \Rightarrow y = 5 or 25 \\ \Rightarrow 5^{\frac{1}{2 x}} = 5 or 25 \\ \Rightarrow x = 1 / 2 or 1 / 4 \end{array}$