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Source: 2023 AMC 12A Problem 19

November 9, 2023
2023 AMC 12AAMCAMC 122023 AMC

Problem Statement

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?
<spanclass=′latex−bold′>(A)</span>(log⁡20237⋅log⁡2023289)2<spanclass=′latex−bold′>(B)</span>log⁡20237⋅log⁡2023289<spanclass=′latex−bold′>(C)</span>1<spanclass=′latex−bold′>(D)</span>log⁡72023⋅log⁡2892023<spanclass=′latex−bold′>(E)</span>(log⁡72023⋅log⁡2892023)2<span class='latex-bold'>(A) </span>(\log_{2023}7 \cdot \log_{2023}289)^2 \qquad<span class='latex-bold'>(B) </span>\log_{2023}7 \cdot \log_{2023}289\qquad<span class='latex-bold'>(C) </span>1\qquad<span class='latex-bold'>(D) </span>\log_{7}2023 \cdot \log_{289}2023\qquad<span class='latex-bold'>(E) </span>(\log_{7}2023 \cdot \log_{289}2023)^2
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