상용로그 가수

$\log a = n + \alpha ( 0 < \alpha < 1 )$ $\displaystyle \log \sqrt[n]{\alpha} = 1 + \frac{\alpha}{n}$ $\displaystyle \therefore \alpha + \frac{\alpha}{n} = 1 ( \because \alpha < 1 , \frac{\alpha}{n} < 1 )$ $\displaystyle \frac{n + 1}{n ^{2} + 8} \log a = 1$ $\displaystyle \log a = \frac{n ^{2} + 8}{n + 1} = n + \alpha \therefore \alpha = \frac{8 - n}{1 + n}$ $\displaystyle \frac{8 - n}{1 + n} + \frac{8 - n}{n + n ^{2}} = 1$ $- n ^{2} + 7 n + 8 = n ^{2} + n$ $n ^{2} - 3 n - 4 = 0$ $\displaystyle \therefore n = 4 ( \because n \text{은} \text{자연수} ) , \log a = \frac{24}{5}$ $\displaystyle \therefore \frac{\log a}{n} = \frac{6}{5}$