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Logarithm Inequality

Source: 2014 AMC 12B #20

February 20, 2014
logarithmsinequalitiesalgebrafunctiondomaincalculusintegration

Problem Statement

For how many positive integers xx is log⁡10(x−40)+log⁡10(60−x)<2\log_{10}{(x-40)} + \log_{10}{(60-x)} < 2?
<spanclass=′latex−bold′>(A)</span> 10<spanclass=′latex−bold′>(B)</span> 18<spanclass=′latex−bold′>(C)</span> 19<spanclass=′latex−bold′>(D)</span> 20<spanclass=′latex−bold′>(E)</span> infinitely many{ <span class='latex-bold'>(A)</span>\ 10\qquad<span class='latex-bold'>(B)</span>\ 18\qquad<span class='latex-bold'>(C)</span>\ 19\qquad<span class='latex-bold'>(D)</span>}\ 20\qquad<span class='latex-bold'>(E)</span>\ \text{infinitely many}
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