Tasty Logarithm #5

$\begin{aligned} \sqrt{1995}x^{\log_{1995}{x}} & = x^2 \\ \log_{1995} {\left(\sqrt{1995}x^{\log_{1995}{x}}\right)} & = \log_{1995} {x^2} \\ \log_{1995} {\sqrt{1995}} + \log_{1995} {x^{\log_{1995}{x}}} & = \log_{1995} {x^2} \\ \frac{1}{2} + \log_{1995} {x}\dot{}\log_{1995} {x} & = 2\log_{1995} {x} \\ \Rightarrow \left( \log_{1995} {x} \right)^2 - 2\log_{1995} {x} + \frac{1}{2} & = 0 \end{aligned}$

There are two roots $x_1$ and $x_2$ , and by Vieta's formulas, we have the sum of roots:

$\log_{1995} {x_1} + \log_{1995} {x_2} = \log_{1995} {x_1x_2} = 2 \\ \Rightarrow x_1x_2 = 1995^2 = 3980025\\ \Rightarrow \left(x_1x_2\right) \mod{1000} = \boxed{25}$