MathDB
Logarithm Terms Are Integers

Source: 2019 AMC 12A #15

February 8, 2019
logarithmsAMCAMC 12AMC 12 A2019 AMC 12A2019 AMC

Problem Statement

Positive real numbers aa and bb have the property that log⁡a+log⁡b+log⁡a+log⁡b=100 \sqrt{\log{a}} + \sqrt{\log{b}} + \log \sqrt{a} + \log \sqrt{b} = 100 and all four terms on the left are positive integers, where log\text{log} denotes the base 10 logarithm. What is abab?
<spanclass=′latex−bold′>(A)</span>1052<spanclass=′latex−bold′>(B)</span>10100<spanclass=′latex−bold′>(C)</span>10144<spanclass=′latex−bold′>(D)</span>10164<spanclass=′latex−bold′>(E)</span>10200<span class='latex-bold'>(A) </span> 10^{52} \qquad <span class='latex-bold'>(B) </span> 10^{100} \qquad <span class='latex-bold'>(C) </span> 10^{144} \qquad <span class='latex-bold'>(D) </span> 10^{164} \qquad <span class='latex-bold'>(E) </span> 10^{200}
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