2022 AIME II Problem 4

There is a positive real number x not equal to either \frac{1}{20} or \frac{1}{2} such that

\log _{20 x}(22 x)=\log _{2 x}(202 x)

The value \log _{20 x}(22 x) can be written as \log _{10}\left(\frac{m}{n}\right), where m and n are relatively prime positive integers. Find m+n.