거듭제곱근 로그

[출제의도] 로그의 성질을 활용하여 문제해결하기 (가)에서 $\displaystyle \sqrt[3]{a}$는 $ab$의 네제곱근이므로 $\displaystyle a ^{\frac{4}{3}} = ab$, $\displaystyle b = a ^{\frac{1}{3}}$ (나)에서 $\log _{a} bc + \log _{b} ac$$\displaystyle = \log _{a} a ^{\frac{1}{3}} c + \log _{a ^{\frac{1}{3}}} ac$ $\displaystyle = \frac{1}{3} \log _{a} a + \log _{a} c + 3 \left( \log _{a} a + \log _{a} c \right)$ $\displaystyle = \frac{10}{3} + 4 \log _{a} c = 4$ $\displaystyle \log _{a} c = \frac{1}{6}$, $\displaystyle c = a ^{\frac{1}{6}}$ 따라서 $\displaystyle a = \left( \frac{b}{c} \right) ^{k} = \left( \frac{a ^{\frac{1}{3}}}{a ^{\frac{1}{6}}} \right) ^{k} = \left( a ^{\frac{1}{3} - \frac{1}{6}} \right) ^{k} = a ^{\frac{k}{6}}$이므로 $k = 6$