지수 로그 관계식

$\displaystyle 2 ^{\frac{1}{x}} = 80$ … ㉠, $\displaystyle 2 ^{\frac{2}{y}} = \frac{1}{10}$ … ㉡, $\displaystyle 2 ^{\frac{1}{z}} = a ^{\frac{1}{3}}$ … ㉢ ㉠$\times$㉡$\div$㉢ 하면 $\displaystyle 2 ^{\frac{1}{x} + \frac{2}{y} - \frac{1}{z}} = \frac{80 \times \frac{1}{10}}{\sqrt[3]{a}}$ $\displaystyle \frac{80 \times \frac{1}{10}}{\sqrt[3]{a}} = 2$, $\displaystyle \sqrt[3]{a} = 4$ $\therefore$ $a = 4 ^{3} = 64$ [다른풀이] $\displaystyle \frac{1}{x} = \log _{2} 80 ,$ $\displaystyle \frac{1}{y} = \log _{4} \frac{1}{10} ,$ $\displaystyle \frac{1}{z} = \log _{8} a$ $\displaystyle \frac{1}{x} + \frac{2}{y} - \frac{1}{z} = \log _{2} \frac{80 \times \frac{1}{10}}{\sqrt[3]{a}} = 1$ $\displaystyle 2 ^{1} = \frac{80 \times \frac{1}{10}}{\sqrt[3]{a}}$, $\displaystyle \sqrt[3]{a} = 4$ $\therefore$ $a = 4 ^{3} = 2 ^{6} = 64$