2022 AIME II Problem 4

There is a positive real number x not equal to either \tfrac{1}{20} or \tfrac{1}{2} such that\log_{20x} (22x)=\log_{2x} (202x).The value \log_{20x} (22x) can be written as \log_{10} (\tfrac{m}{n}), where m and n are relatively prime positive integers. Find m+n.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.