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== 対数計算1 ==(例題→選択問題)
 y=axaとする指数関数という.
例1 y=2x2 を底とする指数関数と呼ばれる.
  (これは2次関数 y=x2 とは全く別のものである.)

y=2x において,与えられた y の値に対応する x の値を表わすために,log2y という記号を導入し, 2とする対数という.
 log2y において,y真数という.

[ 対数の定義 ] ( aとする対数loga yの定義 )
y=axx=loga y

◎[重要]
 この定義により,ある対数が何を表わしているかは,指数の形に直してみれば分かる.

※ 高校では真数は「正の数」だけとする.
y=ax ( y>0 ) だから x=loga y ( y>0 )
例2 8=233=log28
 左辺の式と右辺の式は,どちらに書いてもよい.
8=23log28=3
例3 52=25log525=2
例4 3=log101000103=1000 となることを表わしている.
 《要点》
M=aplogaM=p
logaM=pM=ap
【まとめると】
 指数→対数,対数→指数のどちら向きに変形するときも
底は底,外は中,中は外
=底=log
例題 次の式を指数の形で表せ.4=log216
解答 24=16
42=16 と答えると,式自体は正しくても,この対数を指数の形に直したものになっていないことに注意.

※ 同様にして 8.13n=2 を対数の形に直せば
log82=.13n は○ , log28=3 は×
【問題1】 次の式を指数の形で表せ.
正しい選択肢をクリックしなさい.以下の問題も同様.
解答すれば採点結果が表示され,解説を読むことができます.

(1) log10100=2
⇒ 210=100 , 2100=10 , 102=100 , 10100=2
(2) 5=log232
⇒ 25=32,  232=5,  52=32

532=2,  322=5,  325=2
【問題2】 次の式を対数の形で表せ.
(1) 34=81
⇒ log34=81 ,log381=4 ,log43=81 ,log481=3
(2) 0.1−3=1000
⇒ 1000=log−30.1 , −3=log10000.1

−3=log0.11000 , 3=log101000

 [ 特別な真数 1 , a ]
 a>0 , a1 となるどんな a についても
______a0=1 だから loga1=0 が成立する.
______a1=a だから logaa=1 が成立する.

※前半は「真数が1のときは,底が何であっても対数の値は0になる」ということ.
後半は,「底と真数が同じならば,底が何であっても対数の値は1になる」ということ.
例1 log21=0 , log22=1
例2 log31=0 , log33=1
例3 log101=0 , log1010=1
【問題3】 次の値を求めよ.
(1) log33
⇒ 0 ,  1 ,  3 ,  −1
(2) log51
⇒ 0 ,  1 ,  5 ,  −1

 [ 対数の和→真数は積 ]
 a>0 , a1 , M,N>0 のとき
______logaM+logaN=logaMN が成立する.
______例1 log54+log53=log5(4×3)=log512
______例2 log62+log63=log6(2×3)=log66=1 (※ logaa=1
(解説)
 指数で表わされた数の積については,次の関係が成り立つ.
axay=ax+y
この関係を対数の形で表わぜば上記の性質が証明できる.
すなわち,______ax=M , ay=N とおくと x=logaM , y=logaN
axay=ax+yMN=ax+y
  
logaM+logaN=logaMNx+y=logaMN

※ 注意
足し算がかけ算に等しいということではない.
logaM+logaN ←#等しくない#→ logaM · logaN

「2つの対数の和」は「真数の積の対数」に等しいということ
  logaM+logaN ←○等しい○→ logaMN
【問題4】 次の式の値に等しいものを選べ.
(1) log102+log103
⇒ 1 ,  log105  , log106 ,  log10log103
(2) log123+log124
⇒ 0 ,  1 ,  log127 ,  log12log124

 [ 対数の差→真数は商 ]
 a>0 , a1 , M,N>0 のとき

______logaM−logaN=loga.MNnn が成立する.

___例1 log618−log69=log6.189nn=log62

___例2 log612−log62=log6.122nn=log66=1 (※ logaa=1
(解説)
 指数で表わされた数の商については,次の関係が成り立つ.

.axaynn=ax−y

この関係を対数の形で表わぜば上記の性質が証明できる.
すなわち,______ax=M , ay=N とおくと x=logaM , y=logaN

.axaynn=ax−y.MNnn=ax−y
  
logaM−logaN=loga.MNnnx−y=loga.MNnn
※ 注意
引き算が割り算に等しいということではない.

  logaM−logaN ←#等しくない#→ .logaMlogaNnnnnn

「2つの対数の差」は「真数の商の対数」に等しいということ

  logaM−logaN ←○等しい○→ loga.MNnn
【問題5】 次の式の値に等しいものを選べ.
(1) log310−log32

log35, log38, log312, log320, .log310log32nnnnn

(2) log26−log23
⇒ 0 ,  1 ,  log23 ,  log2log124



 [ 真数のn乗→対数のn倍 ]
 a>0 , a1 , M>0 のとき (n は任意の実数)
__logaMn = n logaM が成立する.

logaMnloga(Mn) を表わす.(logaM)n を表わすときは必ず「かっこ」を付ける.

__例1 log10(23) = 3log102

__例2 log381=log334 = 4log33=4 (※ logaa=1

分数→負の指数,累乗根→分数の指数についても成り立つ.
__例3 log3 .19n=log33−2 = −2log33=−2 (※logaa=1

  引き算の方が得意な人は次のように変形してもよい.
__ log3 .19n=log31 − log39 = 0−2=−2 (※ loga1=0

__例4 log3 .3√ni=log33.12n = .12n log33= .12n(※logaa=1
(解説)
 指数の計算において,(ax)n=ax n が成り立つ.
この関係を対数の形で表わぜば上記の性質が証明できる.
すなわち,__ax=M とおくと
ax=Mx=logaM
 
ax n=MnlogaMn=x n=n logaM
【問題6】 次の式の値に等しいものを選べ.
(1) log281
⇒ 3log24,   4log23,   2log34

4log32,   2log43,   3log42

(2) log4 .116nn
⇒ −2,   −1,   0,   1,   2

(3) log3 .27√nni

− 3,   .32n,   .12n,   .12n,   .32n ,   3


 [ 係数があるとき ]
 a>0 , a1 , M>0 のとき (n は任意の実数)
__n logaM = logaMn

を利用して真数に吸収するとよい.

__例1 3log102= log10(23) = log108

次のような分数も「分数の指数」(=その意味は累乗根)に直せる.

__例2 .log492nnnn = .12nlog49= log4(9.12n) = log43


__例3 2log36 − .log3643nnnnn = log362.13n log364

__= log336 − log364.13n = log336 − log34

__= log3.364nn = log39 = 2

【問題7】 次の式の値に等しいものを選べ.
(1) 2log43
⇒ log45,    log46,    log48,    log49

(2) .log1062nnnnn
⇒ log103,   log1036,   log10.6√ni

log103,   log1036,   log10.6√ni


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■[個別の頁からの質問に対する回答][対数計算1について/18.7.30]
すごくわかりやすかったです ありがとうございました!
=>[作者]:連絡ありがとう.
■[個別の頁からの質問に対する回答][対数計算について/18.5.3]
どのページも説明がわかりやすく、大変参考になります。 [係数があるとき]の例3の第1式の右辺第2項の 底が4となっていますが、正しくは3ではないでしょうか。
=>[作者]:連絡ありがとう.入力ミスですので訂正しました.
■[個別の頁からの質問に対する回答][対数計算1について/18.3.1]
解説の横にそれと似たような問題があって助かります! 私の学校は県で二番目の進学校なのであまり演習をしてくれません。 だから、このサイトは本当にありがたいです。 これからもよろしくお願いします!
=>[作者]:連絡ありがとう.
■[個別の頁からの質問に対する回答][対数計算1について/17.2.12]
制御工学の問題で出てきた,20 log K = G (Gはゲイン[dB])のときのゲイン定数 K の求め方を知りたくて検索しており拝見させていただきましたが,残念ながらこのサイトでは分かりませんでした.デシベルが絡む計算はややこしく,いつも忘れてしまいます(-_-;)
=>[作者]:連絡ありがとう.それぞれの専門分野ごとの約束事があれば別として,高校数学での取り扱いは,その頁の先頭に書いてあります「対数の定義」を見てもらえばできます.

だと思いますが.
■[個別の頁からの質問に対する回答][対数計算について/17.1.8]
この前は、解説ありがとうございました! この前と同じような問題ですが…途中までわかったのですが、その先がわかりません。教えてください! log2(1/9)×log3(1/8) =(-log2 9 )×(-log3 8) この先の計算方法を教えてください! 底の公式にしても前の問題のようにできないのでどうすれば良いのかわかりません…
=>[作者]:連絡ありがとう.今度は底の変換公式の練習になります.
…(底の変換公式)
(ここでcは>0, ≠1なら何でもよいので都合の良いものを選ぶ)

において,(ア)全部の底を10にそろえる場合は


こので「ノミの三段跳び」に持ち込む:

とすると,次のように約分できて

(イ)全部の底を2にそろえる場合は


(ウ)底を3にそろえてもできる

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