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Domain of Composition of Logarithms

Source: 2014 AMC 12A #18

February 5, 2014
algebrafunctiondomainlogarithmsnumber theoryAMC

Problem Statement

The domain of the function f(x)=log⁡12(log⁡4(log⁡14(log⁡16(log⁡116x))))f(x)=\log_{\frac12}(\log_4(\log_{\frac14}(\log_{16}(\log_{\frac1{16}}x)))) is an interval of length mn\tfrac mn, where mm and nn are relatively prime positive integers. What is m+nm+n?
<spanclass=′latex−bold′>(A)</span>19<spanclass=′latex−bold′>(B)</span>31<spanclass=′latex−bold′>(C)</span>271<spanclass=′latex−bold′>(D)</span>319<spanclass=′latex−bold′>(E)</span>511<span class='latex-bold'>(A) </span>19\qquad <span class='latex-bold'>(B) </span>31\qquad <span class='latex-bold'>(C) </span>271\qquad <span class='latex-bold'>(D) </span>319\qquad <span class='latex-bold'>(E) </span>511\qquad
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