Big Summation II

$\begin{aligned} X & = \frac 1{\log_2 2020} + \frac 1{\log_3 2020} + \frac 1{\log_4 2020} + \cdots + \frac 1{\log_{2020} 2020} & \small \blue{\text{Note that }\log_b a = \frac {\log a}{\log b}} \\ & = \frac {\log 2 + \log 3 + \log 4 + \cdots + \log 2020}{\log 2020} & \small \blue{\text{and }\log ab = \log a + \log b} \\ & = \frac {\log 2020!}{\log 2020} \\ & = \boxed{\frac 1{\log_{2020!}2020}} \end{aligned}$

We know that $\boxed{\frac{1}{\log_a {N}} = \log_{N} a}$

Thus, the expression $\frac{1}{\log_2 {2020}} +\frac{1}{\log_3 {2020}} + \frac{1}{\log_4 {2020}}+\dots +\frac{1}{\log_{2020} {2020}}$ can be written as: $= \log_{2020} 2 + \log_{2020} 3+ \dots+ \log_{2020} {2020}$

$=\log_{2020}( 1\times2\times3\times\dots\times2020) = \log_{2020} 2020!$ $\hspace{6cm}\color{#69047E}({\log_a N +\log_a M = \log_a M.N})$

$= \frac {1}{\log_{2020!}{2020}}$