AMC 12 2023a Problem 19

What is the product of all the solutions to the equation $$\log_{7x}2023 \cdot \log_{289x} 2023 = \log_{2023x} 2023?$$

$\textbf{(A) }(\log_{2023}7 \cdot \log_{2023}289)^2 \textbf{(B) }\log_{2023}7 \cdot \log_{2023}289 \textbf{(C) } 1 \\ \\ \textbf{(D) }\log_{7}2023 \cdot \log_{289}2023 \textbf{(E) }(\log_{7}2023 \cdot \log_{289}2023)^2$