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Log Probability

Source: AMC 12 2006B, Problem 20

February 17, 2006
probabilityfloor functionlogarithmsfunctioncalculusintegrationinequalities

Problem Statement

Let x x be chosen at random from the interval (0,1) (0,1). What is the probability that \lfloor\log_{10}4x\rfloor \minus{} \lfloor\log_{10}x\rfloor \equal{} 0? Here ⌊x⌋ \lfloor x\rfloor denotes the greatest integer that is less than or equal to x x.
<spanclass=′latex−bold′>(A)</span>18<spanclass=′latex−bold′>(B)</span>320<spanclass=′latex−bold′>(C)</span>16<spanclass=′latex−bold′>(D)</span>15<spanclass=′latex−bold′>(E)</span>14 <span class='latex-bold'>(A) </span> \frac 18 \qquad <span class='latex-bold'>(B) </span> \frac 3{20} \qquad <span class='latex-bold'>(C) </span> \frac 16 \qquad <span class='latex-bold'>(D) </span> \frac 15 \qquad <span class='latex-bold'>(E) </span> \frac 14
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