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AMC 12A 2023 (Problem 19)

FunctionsHard
What is the product of all the solutions to the equation log⁡7x2023⋅log⁡289x2023=log⁡2023x2023?\log_{7x}2023\cdot\log_{289x}2023=\log_{2023x}2023?
(A)  (log⁡20237⋅log⁡2023289)2\text{(A)}\;(\log_{2023}7\cdot\log_{2023}289)^2(B)  log⁡20237⋅log⁡2023289\text{(B)}\;\log_{2023}7\cdot\log_{2023}289(C)  1\text{(C)}\;1(D)  log⁡72023⋅log⁡2892023\text{(D)}\;\log_72023\cdot\log_{289}2023(E)  (log⁡72023⋅log⁡2892023)2\text{(E)}\;(\log_72023\cdot\log_{289}2023)^2