2020 AMC 12A Problems/Problem 10

Solution

Any logarithm in the form $\log_{a^b} c = \frac{1}{b} \log_a c$ .

so $\log_2{(\log_{2^4}{n})} = \log_{2^2}{(\log_{2^2}{n})}.$

becomes

$\log_2({\frac{1}{4}\log_{2}{n}}) = \frac{1}{2}\log_2({\frac{1}{2}\log_2{n}}).$

Using $\log$ property of addition, we can expand the parentheses into

$\log_2{(\frac{1}{4})}+\log_2{(\log_{2}{n}}) = \frac{1}{2}(\log_2{(\frac{1}{2})} +\log_{2}{(\log_2{n})}).$