Tallying Tiny Logs

Algebra Level 2

$S=\dfrac{2\times2016}{\log_{2}(2016!^{2})}+\dfrac{3\times2016}{\log_{3}(2016!^{3})}+\dfrac{4\times2016}{\log_{4}(2016!^{4})}+\dots+\dfrac{2016\times2016}{\log_{2016}(2016!^{2016})}$

Find $S$ .

Notation: $!$ denotes the factorial notation. For example, $8! = 1\times2\times3\times\cdots\times8$ .