로그 그래프 기울기 합

[출제의도] 로그함수의 성질 이해하기 $\mathrm{P} _{\mathit{n}} ( \mathit{n} , \log _{3} n ) , \mathrm{P} _{\mathit{n} + 1} ( \mathit{n} + 1 , \log _{3} ( n + 1 ) )$ $\displaystyle g ( n ) = \frac{\log _{3} ( n + 1 ) - \log _{3} n}{n + 1 - n} = \log _{3} {\frac{n + 1}{n}}$ $\displaystyle \therefore \sum\limits _{k = 1} ^{80} g ( k ) = \log _{3} \frac{2}{1} + \log _{3} \frac{3}{2} + \cdots + \log _{3} \frac{81}{80}$ $\displaystyle = \log _{3} \left( {\frac{2}{1} \times {\frac{3}{2} \times \cdots \times {\frac{81}{80} )}}} \right.$ $= \log _{3} 81 = 4$