対数の基本公式

by nomura · 2020年4月10日

(1)対数の和

\[ \log M+\log N=\log MN \]

(2)べき乗の対数

\[ \log M^{r}=r\log M \]

(3)底の変換

\[ \log_{a}b=\frac{\log_{c}b}{\log_{c}a} \]

(4)底と真数の交換

\[ \log_{a}b=\frac{1}{\log_{b}a} \]

(1)

\begin{align*} \log M+\log N & =\log\left(\exp(\log M+\log N)\right)\\ & =\log\left(\exp(\log M)\exp(\log N)\right)\\ & =\log\left(MN\right) \end{align*}

(2)

\begin{align*} \log M^{r} & =\log\exp^{r}(\log M)\\ & =\log\exp(r\log M)\\ & =r\log M \end{align*}

(3)

\begin{align*} \log_{a}b & =\log_{a}c^{\log_{c}b}\\ & =\log_{a}c^{\log_{c}a\frac{\log_{c}b}{\log_{c}a}}\\ & =\log_{a}a^{\frac{\log_{c}b}{\log_{c}a}}\\ & =\frac{\log_{c}b}{\log_{c}a} \end{align*}

(4)

\begin{align*} \log_{a}b & =\frac{\log_{b}b}{\log_{b}a}\\ & =\frac{1}{\log_{b}a} \end{align*}