AMC 12A 2021 Spring Problem 14

What is the value of

\left(\sum_{k=1}^{20} \log _{5^{k}} 3^{k^{2}}\right) \cdot\left(\sum_{k=1}^{100} \log _{9^{k}} 25^{k}\right) ?

Answer Choices
A. 21
B. 100 \log _{5} 3
C. 200 \log _{3} 5
D. 2,200
E. 21,000