로그 값의 대소와 거리

$10 < a < 100$인 실수 $a$에 대하여 $1 < \log a < 2$ $\displaystyle \log _{a} 10 = \frac{1}{\log a}$이므로 $\displaystyle \frac{1}{2} < \log _{a} 10 < 1$ $\log 10 a = \log 10 + \log a = 1 + \log a$이므로 $2 < \log 10 a < 3$ $\cdots$$\cdots$ ㉠ $\displaystyle \log \frac{10}{a} = \log 10 - \log a = 1 - \log a$이므로 $\displaystyle - 1 < \log \frac{10}{a} < 0$ $\cdots$$\cdots$ ㉡ $\log _{a} 10 a = \log _{a} 10 + \log _{a} a = \log _{a} 10 + 1$이므로 $\displaystyle \frac{3}{2} < \log _{a} 10 a < 2$ $\cdots$$\cdots$ ㉢ $\displaystyle \log _{a} \frac{a}{10} = \log _{a} a - \log _{a} 10 = 1 - \log _{a} 10$이므로 $\displaystyle 0 < \log _{a} \frac{a}{10} < \frac{1}{2}$ $\cdots$$\cdots$ ㉣ ㉠$\sim$㉣에 의하여 $\displaystyle \log \frac{10}{a} < \log _{a} \frac{a}{10} < \log _{a} 10 a < \log 10 a$이므로 $\displaystyle p = \log \frac{10}{a}$, $\displaystyle q = \log _{a} \frac{a}{10}$, $r = \log _{a} 10 a$, $s = \log 10 a$ $\displaystyle {\overline{\mathrm{PS}}}$$= s - p$$\displaystyle = \log 10 a - \log \frac{10}{a}$ $= \left( 1 + \log a \right) - \left( 1 - \log a \right) = 2 \log a$ $\displaystyle = \frac{10}{3}$ $\displaystyle \log a = \frac{5}{3}$ $\displaystyle {\overline{\mathrm{QR}}}$$= r - q$$\displaystyle = \log _{a} 10 a - \log _{a} \frac{a}{10}$ $= \left( \log _{a} 10 + 1 \right) - \left( 1 - \log _{a} 10 \right)$ $\displaystyle = 2 \log _{a} 10 = \frac{2}{\log a} = \frac{6}{5}$ 따라서 $\displaystyle 30 \times \overline{{\mathrm{QR}}} = 30 \times \frac{6}{5} = 36$