로그와 지수 계산

$\displaystyle a = \log _{9} \left( 7 - 4 \sqrt{3} \right)$이므로 $\displaystyle 9 ^{a} = 3 ^{2a} = ( 3 ^{a} ) ^{2} = 7 - 4 \sqrt{3}$ $\therefore 3 ^{a}$$\displaystyle = \sqrt{{7 - 4 \sqrt{3}}}$$\displaystyle = \sqrt{7 - 2 \sqrt{12}}$$\displaystyle = \sqrt{( 2 - \sqrt{3} ) ^{2}}$$\displaystyle = 2 - \sqrt{3}$ $3 ^{-a}$$\displaystyle = \frac{1}{2 - \sqrt{3}} = 2 + \sqrt{3}$ $\therefore 3 ^{a} + 3 ^{-a} = 4$ [다른 풀이] $\displaystyle 9 ^{a} = 7 - 4 \sqrt{3}$, $\displaystyle 9 ^{-a} = \frac{1}{7 - 4 \sqrt{3}} = 7 + 4 \sqrt{3}$이므로 $( 3 ^{a} + 3 ^{-a} ) ^{2} = 9 ^{a} + 2 + 9 ^{-a}$$= 16$ $\therefore 3 ^{a} + 3 ^{-a} = 4$ (∵ $3 ^{a} > 0$)