Solution:

By the change-of-base formula, $$\log_{2^x} 3^{20} = \frac{20 \log 3}{x \log 2} \qquad \text{and} \qquad \log_{2^{x+3}} 3^{2020} = \frac{2020 \log 3}{(x + 3) \log 2}.$$ Cancelling the common factor $\frac{\log 3}{\log 2}$ leaves $\frac{20}{x} = \frac{2020}{x + 3}.$

Cross-multiplying gives $20x + 60 = 2020x,$ so $2000x = 60$ and $x = \frac{3}{100}.$ Thus $m + n = 3 + 100 = 103.$