2014 AMC 12A Problem 21

For every real number x, let \lfloor x\rfloor denote the greatest integer not exceeding x, and let f(x)=\lfloor x\rfloor(2014^{x-\lfloor x\rfloor}-1). The set of all numbers x such that 1\leq x<2014 and f(x)\leq 1 is a union of disjoint intervals. What is the sum of the lengths of those intervals?

\textbf{(A) }1\qquad \textbf{(B) }\dfrac{\log 2015}{\log 2014}\qquad \textbf{(C) }\dfrac{\log 2014}{\log 2013}\qquad \textbf{(D) }\dfrac{2014}{2013}\qquad \textbf{(E) }2014^{\frac1{2014}}\qquad

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