2023 AMC 12A Problem 19

What is the product of all the solutions to the equation \[ \begin{gathered} \log_{7x}2023\cdot\log_{289x}2023\\ {}=\log_{2023x}2023? \end{gathered} \]

\textbf{(A)} (\log_{2023}7\cdot\log_{2023}289)^2 \qquad \textbf{(B)} \log_{2023}7\cdot\log_{2023}289 \qquad \textbf{(C)} 1 \qquad \textbf{(D)} \log_7 2023\cdot\log_{289}2023 \qquad \textbf{(E)} (\log_7 2023\cdot\log_{289}2023)^2

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.