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指数不等式の解法 | 夢を叶える塾
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指数不等式の問題の解法

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指数関数の性質を利用して不等式の解を求める問題です。

基本問題

次の不等式の解を求めなさい。
(1)\displaystyle \left(\frac{1}{3}\right)^x \leq 3
(2)2\cdot 3^x+3^{2x}\leq 15

解き方

単純な不等式であれば、グラフを思い出しましょう。
なお底は1より大きい数で考えるとわかりやすいと思います。
方程式の解き方は指数方程式の問題の解法が参考になります。
二次不等式の解き方は二次不等式の解法が参考になります。

解説

(1)\displaystyle \left(\frac{1}{3}\right)^x \leq 3
底をそろえた方がいいですね。
このとき、2つのそろえ方があります。
それぞれ解説していきますね。

まず3でそろえてみましょう。
\displaystyle \left(\frac{1}{3}\right)^x \leq 3
\displaystyle \left(3^{-1}\right)^x \leq 3^1
\displaystyle 3^{-x} \leq 3^1
底が1より大きいですね。
このときグラフは右肩上がりです。
指数部分が大きければ大きいほど大きくなります。
つまり指数部分の大小関係と、指数を計算した結果の大小関係が変わらないという事です。
これが底は1より大きい数で考えたほうがわかりやすい理由です。
\displaystyle -x \leq 1
\displaystyle x \geq -1
答えです。

次に\displaystyle \frac{1}{3}でそろえてみましょう。
\displaystyle \left(\frac{1}{3}\right)^x \leq 3
\displaystyle \left(\frac{1}{3}\right)^x \leq \left(\frac{1}{3}\right)^{-1}
底が1より小さいですね。
このときグラフは右肩下がりです。
指数部分が大きければ大きいほど小さくなります。
つまり指数部分の大小関係と、指数を計算した結果の大小関係が反転するという事です。
指数の不等式に変えるときに不等号を反転させる必要があります。
\displaystyle x \geq -1
答えです。

なお、求めた答えが正しいか確認しておいた方が良いでしょう。
x=0,x=-2あたりを使って、
x=0のとき\displaystyle 1 \leq 3で確かに正しい
x=-2のとき\displaystyle 9 \leq 3で確かに正しくない

(2)2\cdot 3^x+3^{2x}\leq 15
指数方程式のときと同様、t=3^xで置換します。
2\cdot (3^x)+(3^x)^2 \leq 15
2t+t^2\leq 15
二次不等式ですね。

二次不等式を解いていきます。
t^2+2t-15\leq 0
(t-3)(t+5)\leq 0
左辺の二次関数のグラフは下に凸でt=3,-5で解を持っていますね。
-5 \leq t \leq 3
指数を置換する際に注意すべき点がありましたね。
t=3^xで置換したので、0< tとなります
これらを満たすtが不等式の解になります。
従って共通部分をとって0 < t \leq 3が答えです。

終わりに

単純な不等式の場合はグラフを思い出してください。
二次不等式まで持っていけない場合は指数方程式の問題の解法を、二次不等式から解けない場合は二次不等式の解法をそれぞれ参考にしてみてください。

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