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Logarithms With Powers of 2

Source: 2019 AMC 12A #12

February 8, 2019
AMCAMC 12AMC 12 Alogarithmsalgebraquadratic formula2019 AMC 12A

Problem Statement

Positive real numbers x≠1x \neq 1 and y≠1y \neq 1 satisfy log⁡2x=log⁡y16\log_2{x} = \log_y{16} and xy=64xy = 64. What is (log⁡2xy)2(\log_2{\tfrac{x}{y}})^2?
<spanclass=′latex−bold′>(A)</span>252<spanclass=′latex−bold′>(B)</span>20<spanclass=′latex−bold′>(C)</span>452<spanclass=′latex−bold′>(D)</span>25<spanclass=′latex−bold′>(E)</span>32<span class='latex-bold'>(A) </span> \frac{25}{2} \qquad<span class='latex-bold'>(B) </span> 20 \qquad<span class='latex-bold'>(C) </span> \frac{45}{2} \qquad<span class='latex-bold'>(D) </span> 25 \qquad<span class='latex-bold'>(E) </span> 32
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