AMC 12 2023 A Problem 19

What is the product of all solutions to the equation

$$\log _{7 x} 2023 \cdot \log _{289 x} 2023=\log _{2023 x} 2023?$$

Answer Choices
A. \left(\log _{2023} 7 \cdot \log _{2023} 289\right)^{2}
B. \log _{2023} 7 \cdot \log _{2023} 289
C. 1
D. \log _{7} 2023 \cdot \log _{289} 2023
E. \left(\log _{7} 2023 \cdot \log _{289} 2023\right)^{2}