AMC 12 2006b 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 $x$.

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