2006 AMC 12B Problem 20

Let x be chosen at random from the interval (0,1). What is the probability that \lfloor\log_{10}4x\rfloor - \lfloor\log_{10}x\rfloor = 0? Here \lfloor x\rfloor denotes the greatest integer that is less than or equal to x.

\mathrm{(A)}\ \frac 18 \qquad \mathrm{(B)}\ \frac 3{20} \qquad \mathrm{(C)}\ \frac 16 \qquad \mathrm{(D)}\ \frac 15 \qquad \mathrm{(E)}\ \frac 14

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.