2006 AMC 12B Problem 20

The equation says $\lfloor \log_{10} x \rfloor = \lfloor \log_{10} 4x \rfloor,$ i.e. $x$ and $4x$ lie in the same interval $[10^n, 10^{n+1}).$ This holds exactly when $10^n \le x$ and $4x \lt 10^{n+1},$ that is $10^n \le x \lt \dfrac{10^{n+1}}{4}.$ Within $[10^n, 10^{n+1}),$ the favorable fraction is $$\begin{aligned} &\frac{\frac{10^{n+1}}{4} - 10^n}{10^{n+1} - 10^n} \\ &= \frac{\frac{10}{4} - 1}{10 - 1} = \frac{1}{6}. \end{aligned}$$ Since this fraction is the same on every such interval, the overall probability is $\dfrac{1}{6}.$ Thus, the correct answer is C.