Chọn phương án B.

Điều kiện: \(\begin{cases}
x>0\\ x\neq1.
\end{cases}\)

Ta có$$\begin{eqnarray*}
&\log_x(125x)\cdot\log_{25}^2x&=1\\
\Leftrightarrow&\left(\log_x125+\log_xx\right)\cdot\left(\log_{25}x\right)^2&=1\\
\Leftrightarrow&\left(\log_x5^3+1\right)\cdot\left(\log_{5^2}x\right)^2&=1\\
\Leftrightarrow&\left(3\log_x5+1\right)\cdot\left(\dfrac{1}{2}\log_5x\right)^2&=1\\
\Leftrightarrow&\dfrac{3\log_x5+1}{4\left(\log_x5\right)^2}&=1\\
\Leftrightarrow&\dfrac{3\log_x5+1}{4\left(\log_x5\right)^2}-1&=0\\
\Leftrightarrow&-4\left(\log_x5\right)^2+3\log_x5+1&=0\\
\Leftrightarrow&\left[\begin{array}{l}\log_x5=1\\ \log_x5=-\dfrac{1}{4}\end{array}\right.\Leftrightarrow\left[\begin{array}{l}\log_5x=1\\ \log_5x=-4\end{array}\right.\\
\Leftrightarrow&\left[\begin{array}{l}x=5\\ x=5^{-4}\end{array}\right.
\end{eqnarray*}$$
Khi đó, tích các nghiệm cần tìm là $$5\cdot5^{-4}=5^{-3}=\dfrac{1}{125}$$