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対数の解法 | 夢を叶える塾
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対数の問題の解法

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対数の性質を利用して計算する問題です。

基本問題

次の式を簡単にしなさい。
(1)\log_2{20}+\log_2{6}-\log_2{15}
(2)\displaystyle 4\log_2{6}-2\log_2{9}+\frac{1}{2}
(3)\displaystyle \log_2{3}\cdot \log_3{6} \cdot \log_6{4}
(4)\displaystyle (2\log_2{3}+ \log_4{3}) \cdot \log_3{4}

解き方

対数の性質を使って計算を進めます。

底が異なって計算が進められない場合、底の変換公式を使うと底を入れ替えることができます。
\log_a{b}\cdot \log_c{d}=\log_c{b}\cdot \log_a{d}
定期試験で使うときは証明してから使わないと減点される可能性もあるので、事前に確認しておくと良いかもしれません
\displaystyle \log_a{b}\cdot \log_c{d}=\frac{\log_e{b}}{\log_e{a}}\cdot \frac{\log_e{d}}{\log_e{c}}=\frac{\log_e{b}}{\log_e{c}}\cdot \frac{\log_e{d}}{\log_e{a}}=\log_c{b}\cdot \log_a{d}[

解説

(1)\log_2{20}+\log_2{6}-\log_2{15}
底がそろっている場合、対数の足し算は中身を掛け算にし、引き算は割り算ですね。

\displaystyle \log_2{20}+\log_2{6}-\log_2{15}=\log_2{\frac{20\cdot 6}{15}}=\log_2{8}
\log_a{b^c}=c\log_a{b}を使いましょう。
\displaystyle =\log_2{8}=\log_2{2^3}=3\log_2{2}
\log_a{a}=1を使いましょう。
\displaystyle =3\log_2{2}==3
答えです。

(2)\displaystyle 4\log_2{6}-2\log_2{9}+\frac{1}{2}
1つだけ対数が付いていない項が少し計算し難さを出しているので、仲間入りして頂きましょう
\displaystyle \frac{1}{2}=\frac{1}{2}\log_2{2}
それでは計算を進めましょう。
まず単純な足し算引き算で計算ができるので、係数には肩に乗っていただきます
\displaystyle 4\log_2{6}-2\log_2{9}+\frac{1}{2}\log_2{2}=\log_2{6^4}-\log_2{9^2}+\log_2{2^{\frac{1}{2}}}
後は掛け算割り算に直して計算を進めます。
\displaystyle =\log_2{6^4}-\log_2{9^2}+\log_2{2^{\frac{1}{2}}}=\log_2{\frac{2^43^42^{\frac{1}{2}}}{3^4}}=\log_2{2^{\frac{9}{2}}}=\frac{9}{2}\log_2{2}=\frac{9}{2}
答えです。

(3)\displaystyle \log_2{3}\cdot \log_3{6} \cdot \log_6{4}
対数が積となっている場合、底の変換公式が有効ですね。
底を合わせるように変換します。
今回は3にしてみました。
\displaystyle \log_2{3}\cdot \log_3{6} \cdot \log_6{4}=\frac{\log_3{3}}{\log_3{2}}\cdot \log_3{6} \cdot \frac{\log_3{4}}{\log_3{6}}=\frac{1}{\log_3{2}}\cdot \log_3{6} \cdot \frac{\log_3{2^2}}{\log_3{6}}=\frac{\log_3{2^2}}{\log_3{2}}
\displaystyle =\frac{\log_3{2^2}}{\log_3{2}}=\frac{2\log_3{2}}{\log_3{2}}=2
答えです。

もう1つ、解き方に記載している底の入れ替えを使ってみましょう。
\log_a{b}\cdot \log_c{d}=\log_c{b}\cdot \log_a{d}
ですね。
積の形であれば、好きなように入れ替えることができます
\displaystyle \log_2{3}\cdot \log_3{6} \cdot \log_6{4}=\log_3{3}\cdot \log_6{6} \cdot \log_2{4}=\log_2{2^2}=2\log_2{2}=2
答えです。

(4)\displaystyle (2\log_2{3}+ \log_4{3}) \cdot \log_3{4}
どう進めてもよさそうです。
()を外してみます。
\displaystyle (2\log_2{3}+ \log_4{3}) \cdot \log_3{4}=(\log_2{3^2}+ \log_4{3}) \cdot \log_3{2^2}=\log_2{3^2}\cdot \log_3{2^2}+ \log_4{3}\cdot \log_3{2^2}
底を入れ替えると簡単になりそうですね。
\displaystyle =\log_2{3^2}\cdot \log_3{2^2}+ \log_4{3}\cdot \log_3{2^2} =\log_3{3^2}\cdot \log_2{2^2}+ \log_3{3}\cdot \log_4{2^2}=2\log_3{3}\cdot 2\log_2{2}+ 1\cdot 1=4+1=5
答えです。

終わりに

計算問題ではあまり意識しなくても良いかもしれませんが、対数の値がどのようになるかは少しわかりにくいですよね。
\log_2{8}=3あたりを覚えておく(作る)と、2^3=8と比較してわかりやすいのかなと思います
対数の計算の基礎となる公式
\displaystyle \log_a{a}=1
\displaystyle \log_a{b^c}=c\log_a{b}
\displaystyle \log_a{b}+\log_a{c}=\log_a{bc}
\displaystyle \log_a{b}=\frac{log_c{b}}{log_c{a}}
は必ず使いこなせるようになってください。
あと、底の変換公式から得られる底の入替の公式は便利です。

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