Logarithms and 2018!

$\begin{aligned} S & = \frac 1{\log_2 n} + \frac 1{\log_3 n} + \frac 1{\log_4 n} + \cdots + \frac 1{\log_{2018} n} & \small \color{#3D99F6} \text{where }n = 2018! \\ & = \sum_{k=2}^{2018} \frac 1{\log_k n} \\ & = \sum_{k=2}^{2018} \frac {\log k}{\log n} \\ & = \frac 1{\log 2018!} \sum_{k=2}^{2018} \log k \\ & = \frac 1{\sum_{k=2}^{2018} \log k} \sum_{k=2}^{2018} \log k \\ & = \boxed{1} \end{aligned}$