[tex]log_{100} \ 20= \frac{log_2 \ 20}{log_2 \ 100}= \frac{log_2 \ 2^2*5}{log_2 \ 2^2*5^2}= \\\\ = \frac{log_2 \ 2^2 + log_2 \ 5}{log_2 \ 2^2 + log_2 \ 5^2}= \frac{2+log_2 \ 5}{2+ 2 log_2 \ 5} \\\\\\ a \longrightarrow log_2 \ 5 \\\\\\ \boxed{log_{100} \ 20 = \frac{2+a}{2+2a}} [/tex]
