로그함수 그래프와 평행

$a = \log _{4} ( - x )$에서 $x _{1} = - 4 ^{a}$ $a = 2 - \log _{2} x$에서 $x _{2} = 2 ^{2-a}$ $\displaystyle \left| \frac{x _{2}}{x _{1}} \right| = \frac{1}{2}$이므로 $2 ^{2-3a} = 2 ^{-1}$, $a = 1$ $\therefore$$\mathrm{A} \mathit{\left( - 4 , 1 \right)}$, $\mathrm{B} \mathit{\left( 2 , 1 \right)}$

$b = \log _{4} ( - x )$에서 $x _{3} = - 4 ^{b}$ $b = 2 - \log _{2} x$에서 $x _{4} = 2 ^{2-b}$ $\therefore$$\mathrm{C} \mathit{\left( - 4 ^{b} , b \right)}$, $\mathrm{D} \mathit{\left( 2 ^{2 - b} , b \right)}$ 두 직선 $\mathrm{AC}$와 $\mathrm{BD}$가 서로 평행하므로 $\displaystyle \frac{1 - b}{- 4 + 4 ^{b}} = \frac{1 - b}{2 - 2 ^{2-b}}$ $1 - b \neq 0$이므로 $- 4 + 4 ^{b} = 2 - 2 ^{2-b}$ 양변에 $2 ^{b}$을 곱한 후 정리하면 $2 ^{3b} - 6 \times 2 ^{b} + 4 = 0$ $2 ^{b} = X$$\left( X > 0 , X \ne 2 \right)$로 놓으면 $X ^{3} - 6 X + 4 = 0$. $\left( X - 2 \right) \left( X ^{2} + 2 X - 2 \right) = 0$ $\therefore$$\displaystyle X = \mathit{-} 1 + \sqrt{3}$ 따라서 $\displaystyle 2 ^{b} = \mathit{-} 1 + \sqrt{3}$이므로 $\displaystyle \left| \frac{x _{4}}{x _{3}} \right|$$\displaystyle = \left| \frac{2 ^{2-b}}{- 4 ^{b}} \right|$$= 2 ^{2-3b}$$\displaystyle = 4 \left( \frac{1}{2 ^{b}} \right) ^{3}$$\displaystyle = 4 \left( \frac{\sqrt{3} + 1}{2} \right) ^{3}$ $\displaystyle = 5 + 3 \sqrt{3}$