AMC 12 2013a Problem 21

Consider $A = \log (2013 + \log (2012 + \log (2011 + \log (\cdots + \log (3 + \log 2) \cdots ))))$. Which of the following intervals contains $A$?

\textbf{(A)} \ (\log 2016, \log 2017)$$\textbf{(B)} \ (\log 2017, \log 2018)$$\textbf{(C)} \ (\log 2018, \log 2019)$$\textbf{(D)} \ (\log 2019, \log 2020)$$\textbf{(E)} \ (\log 2020, \log 2021)