2013 AMC 12A 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)

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