Logarithms in 2020

$\begin{aligned} \log_{a}{x} + \log_{a}{y} &= \log_{a}{2020} \\ \log_{a}{(x\cdot y)} &= \log_{a}{2020} \\ xy &= 2020 \\ \\ 2020 &= 2^{2} \cdot 5^{1} \cdot 101^{1} \\ \text{Factors of 2020} &= (2+1)(1+1)(1+1) \\ &= \boxed{12} \end{aligned}$

$\text{Since each factor corresponds to another factor to multiply to 2020, there are 12 pairs in total}$

$\begin{array}{c|c|c} &x&y \\ \hline \\ 1.&1&2020 \\ 2.&2&1010 \\ 3.&4&505 \\ 4.&5&404 \\ 5.&10&202 \\ 6.&20&101 \\ 7.&101&20 \\ 8.&202&10 \\ 9.&404&5 \\ 10.&505&4 \\ 11.&1010&2 \\ 12.&2020&1 \end{array}$