Logs with Factorials!

Animated video solution: can be found here

Property References:

$\color{#3D99F6}{1.}$ $\log {(ab)} = \log {(a)} + \log {(b)} \\ \color{#3D99F6}{2.}$ $\log_{a} {b} = \frac{1}{\log_{b} {a}}$

Evaluating the expression

$\begin{aligned} &= \frac{1}{\log _{2} {(n!)}} + \frac{1}{\log _{3} {(n!)}} + \frac{1}{\log _{4} {(n!)}} + \ldots + \frac{1}{\log _{n} {(n!)}} \\ \\ &= {\log _{n!} {(2)}} + {\log _{n!} {(3)}} + {\log _{n!} {(4)}} + \ldots + {\log _{n!} {(n)}} \quad [{\color{#3D99F6}{2}}] \\ \\ &= {\log _{n!} {(2 \cdot 3 \cdot 4 \cdot 5 \cdots n)}} \quad [{\color{#3D99F6}{1}}] \\ \\ &= {\log _{n!} {(n!)}} \\ \\ &= \boxed{1} \end{aligned}$

$\displaystyle \frac{1}{\log_2 (n!)} + \frac{1}{\log_3 (n!)} + \frac{1}{\log_4 (n!)} + ... + \frac{1}{\log_n (n!)}$

$= \displaystyle \frac{1}{\frac{\log (n!)}{\log 2}} + \frac{1}{\frac{\log (n!)}{\log 3}} + \frac{1}{\frac{\log (n!)}{\log 4}} + ... + \frac{1}{\frac{\log (n!)}{\log n}}$

$= \displaystyle \frac{\log 2}{\log n!} + \frac{\log 3}{\log n!} + \frac{\log 4}{\log n!} + ... + \frac{\log n}{\log n!}$

$= \displaystyle \frac{\log 2 + \log 3 + \log 4 + ... + \log n}{\log n!}$

$= \displaystyle \frac{\log 2 \cdot 3 \cdot 4 \cdot ... \cdot n}{\log n!}$

$= \displaystyle \frac{\log n!}{\log n!}$

$= \displaystyle \boxed{1}$