AMC 12 2014a 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?

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