I recently took a math test that had the following problem: $$ \frac{1}{\log_{2}50!} + \frac{1}{\log_{3}50!} + \frac{1}{\log_{4}50!} + \dots + \frac{1}{\log_{50}50!} $$ The sum is equal to 1. I understand that the logs can be broken down into (first fraction shown) $$ \frac{1}{\log_{2}1 + \log_{2}2 + \log_{2}3 + \dots + \log_{2}50} $$
How do the fractions with such irrational values become $1$? Is there a formula or does one simply need to combine fractions and use the basic properties of logs?