対数不等式の問題

■問題

次の対数不等式を解け

${\displaystyle {log}_{3}9\cdot {log}_{\frac{1}{3}}4\leqq {log}_{9}x\leqq {log}_{\frac{1}{3}}4+{log}_{3}2}$

■答

${\displaystyle \frac{1}{256}\leqq x\leqq \frac{1}{4}}$

■解説

${\displaystyle {log}_{3}9{log}_{\frac{1}{3}}4}$${\displaystyle \mathrm{}\leqq {log}_{9}x}$${\displaystyle \leqq {log}_{\frac{1}{3}}4+{log}_{3}2}$

底の変換公式を用いて，底を$3$ にそろえる．その後，対数計算の基本を利用して式を簡単にする．

ここで

${\displaystyle {log}_{3}9}$${\displaystyle ={log}_3{3}^2}$${\displaystyle =2{\log }_{3}3}$${\displaystyle =2}$

${\displaystyle {log}_{\frac{1}{3}}4}$${\displaystyle =\frac{{log}_{3}4}{{log}_3\frac{1}{3}}}$${\displaystyle =-{log}_{3}4}$${\displaystyle ={log}_3\frac{1}{4}}$

${\displaystyle {log}_{9}x}$${\displaystyle =\frac{{log}_{3}x}{{log}_{3}9}}$${\displaystyle =\frac{1}{2}{log}_{3}x}$

よって

${\displaystyle 2\frac{{log}_{3}4}{{log}_3\frac{1}{3}}}$${\displaystyle \leqq \frac{{log}_{3}x}{{log}_{3}9}}$${\displaystyle \leqq \frac{{log}_{3}4}{{log}_3\frac{1}{3}}+{log}_{3}2}$

${\displaystyle 2\left(-{log}_{3}4\right)}$${\displaystyle \leqq \frac{1}{2}{log}_{3}x}$${\displaystyle \leqq -{log}_{3}4+{log}_{3}2}$

さらに，ここで

${\displaystyle 2\left(-{log}_{3}4\right)}$${\displaystyle =2{log}_3\frac{1}{4}}$${\displaystyle ={log}_3{\left(\frac{1}{4}\right)}^2}$${\displaystyle ={log}_3\frac{1}{16}}$

${\displaystyle -{\log }_{3}4+{\log }_{3}2}$${\displaystyle ={log}_3\frac{1}{4}+{log}_{3}2}$${\displaystyle ={log}_3\frac{1}{2}}$

${\displaystyle =\frac{1}{2}{log}_{3}x}$${\displaystyle ={log}_3{x}^{\frac{1}{2}}}$${\displaystyle ={log}_3\sqrt{x}}$

よって

${\displaystyle 2{log}_3\frac{1}{4}}$${\displaystyle \leqq {log}_3{x}^{\frac{1}{2}}}$${\displaystyle \leqq {log}_3\frac{1}{4}+{log}_{3}2}$

${\displaystyle {log}_3\frac{1}{16}}$${\displaystyle \leqq {\log }_3\sqrt{x}}$ ${\displaystyle \leqq {log}_3\frac{1}{2}}$

底が$3$で$1$ より大きい．よって，真数の大小関係は以下のようになる．

${\displaystyle \frac{1}{16}}$${\displaystyle \leqq \sqrt{x}}$${\displaystyle \leqq \frac{1}{2}}$

${\displaystyle \frac{1}{256}}$${\displaystyle \leqq x}$${\displaystyle \leqq \frac{1}{4}}$

■問題へ戻る

ホーム>>カテゴリー分類>>指数／対数>>対数に関する問題>>対数不等式の問題

作成：学生スタッフ

最終更新日： 2023年11月28日