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== 対数関数のグラフ ==
≪要点≫
(A) y=axのグラフを対称移動してy=logaxのグラフを作る方法
 対数の形で書かれる式y=logaxは,指数の形で書かれる式y=axx座標とy座標の役割を入れ替えたものです.
(ただし,a>0, a≠1, x>0の範囲で考えます)
y=ax
xyを入れ替える)
y=logax
x座標とy座標を入れ替えると,「縦のもの」と「横のもの」が入れ替わります.
◎対数関数のグラフy=logaxのグラフは,指数関数のグラフy=axy=xの直線に関して対称移動したものになります.
【例1】
y=2x12=211=202=214=22
(x, y)(1,12)(0, 1)(1, 2)(2, 4)

y=log2x1=log2120=log211=log222=log24
(x, y)(12,1)(1, 0)(2, 1)(4, 2)

【重要】
1. 指数関数の定義域:−∞<x<∞, 値域:0<y<∞

対数関数の定義域:0<x<∞, 値域:−∞<y<∞

2. 指数関数のグラフは常に点(0, 1)を通る

対数関数のグラフは常に点(1, 0)を通る

3. 各々のa (>0)について,指数関数のグラフは点(1, a)を通る

各々のa (>0)について,対数関数のグラフは点(a, 1)を通る

4. 指数関数のグラフは,x軸を漸近線とする

対数関数のグラフはy軸を漸近線とする

5. y=axaを指数関数の底(てい : base)という.
(1) 底a>1のとき,指数関数y=axは単調増加関数になる.
a>1のとき,対数関数y=logaxは単調増加関数になる
【例2】
y=(12)x2=(12)11=(12)012=(12)114=(12)2
(x, y)(1,2)(0, 1)(1,12)(2,14)

y=log12x1=log1220=log1211=log12122=log1214
(x, y)(2,1)(1, 0)(12,1)(14,2)

【重要】
1.4.までは,【例1】と全く同じ
5.の次の点だけが異なる
(2) 底a<1のとき,指数関数y=axは単調減少関数になる.

≪要点≫
(B) 対数logaxの値を使って,xとyの対応表を作ってグラフにする方法
【例3】
y=log3xのグラフを描くには,次のような対応表を作って,滑らかな線で結べばよい
x013139
log3x×−1012
【例4】
y=log13xのグラフを描くには,次のような対応表を作って,滑らかな線で結べばよい
x01913139
log13x×210−1−2

≪要点≫
(C) 2つの対数関数のグラフを重ねて書くときは,底の大小関係と区間0<0, 1<∞に応じて,上下左右をはっきりさせます
【例5】
アのx<0, 0<y<1において3xが下だから
ウの0<x<1, y<0において,log3xは左にある

イのx>0, y>1において3xが上だから
エのx>1, y>0において,log3xは右にある

【グラフを用いて対数の大小比較の問題を解くには(1)】
(1) 右図のように,底a>1のとき,増加関数になるので対数logap, logaqの大小は,真数p, qの大小と一致します.
(2) 底0<a<1のとき,減少関数になるので対数logap, logaqの大小は,真数p, qの大小と逆になります.
【例題1】
次の数を小さいものから順に並べてください.
 log43,1,log413
(解答)
log43, 1=log44, log413において,真数の大小は
13<3<4
ここで,log4xは底が1よりも大きいから増加関数
ゆえに,log413<log43<1・・・(答)
【問題1】
次の数を小さいものから順に並べてください.
 2log23, log213, 2
解答を見る

【例題2】
次の数を小さいものから順に並べてください.
 2log132, log1312, 1
(解答)
2log132=log134
log1312=log132
1=log133
これらの真数は
2<3<4
ここで,log13xは底が1よりも小さいから減少関数
ゆえに,2log132<1<log1312・・・(答)
【問題2】
次の数を小さいものから順に並べてください.
 2, log123, 2log123
解答を見る

【グラフを用いて対数の大小比較の問題を解くには(2)】
1. 底が1よりも大きな異なる数のとき,
(1) 右図アイのように,真数が等しくて1よりも大きいとき
log33<log23のように,底が小さい方の対数が大きくなります.
(2) 右図ウエのように,真数が等しくて1よりも小さいとき
log212<log312のように,底が大きい方の対数が大きくなります.
(3) 右図オアのように,真数も底も異なるとき
log32<log22=log33<log23のように,値が等しい対数を2通りに書いて接続します.

2. 底が1よりも小さな異なる数のとき,
 右図のように,同様に考えることができますが,結果については各自でグラフを描いて判断できるようにしましょう.
3. また,底が1よりも大きな数と小さな数に分かれている場合についても,同様に各自でグラフを描いて判断できるようにしましょう.

【例題3】
次の数を小さいものから順に並べてください.
 1,log43,log45,log35
(解答)
x>1かつ(1<)a<bのとき,logbx<logaxだから
(0<)log45<log35
0<x<1かつ(1<)a<bのとき,logax<logbxだから
1=log33=log313<log413=log43(<0)
1<log43<log45<log55・・・(答)
後半の比較は
1=log414<log413=log43(<0)
でもよい
【問題3】
次の数を小さいものから順に並べてください.
 log1413,log1314,1
解答を見る

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