로그 합 빈칸

[출제의도] 로그의 성질 추론하기 자연수 $n$에 대하여 $\displaystyle \log _{n + 1} \left( n + 2 \right) = \frac{{\square {{}^{{}^{{}^{}}} \log _{2} \left( n + 2 \right) _{{}_{{}_{}}}}}}{\log _{2} \left( n + 1 \right)}$이므로 $\displaystyle \sum\limits _{k = 1} ^{14} \log _{2} \left\{ \log _{k + 1} \left( k + 2 \right) \right\}$ $\displaystyle = \log _{2} \left( \frac{\log _{2} 3}{\log _{2} 2} \right) + \log _{2} \left( \frac{\log _{2} 4}{\log _{2} 3} \right) + \cdots + \log _{2} \left( \frac{\log _{2} 16}{\log _{2} 15} \right)$ $\displaystyle = \log _{2} \left( \frac{\log _{2} 3}{\log _{2} 2} \times \frac{\log _{2} 4}{\log _{2} 3} \times \cdots \times \frac{\log _{2} 16}{\log _{2} 15} \right)$ $\displaystyle = \log _{2} \left( \frac{\left. {\square {{}^{{}^{{}^{}}} \log _{2} 16 _{{}_{{}_{}}}}} \right.}{\log _{2} 2} \right) = \log _{2} 4 = {\square {{}^{{}^{{}^{}}} 2 _{{}_{{}_{}}}}}$ $\therefore$ $f \left( n \right) = \log _{2} \left( n + 2 \right)$, $p = 4$, $q = 2$ 따라서 $f \left( p + q \right) = \log _{2} 8 = 3$