対数関数
任意の正の数\(N\)に対して
\[
N = a^x
\]
を満たすただ 1 つの\(x\)の値を
\[
x = \mathrm{log}_a{N}
\]
と表し、これを\(a\)を底とする\(N\)の対数という。また\(N\)をこの対数の真数という。
対数の性質
\(a>0\)、\(a\neq1\)、\(M>0\)、\(N>0\)で\(r\)が実数のとき以下が成り立つ。
\[
\begin{align}
\mathrm{log}_aMN &= \mathrm{log}_aM + \mathrm{log}_aN \\
\mathrm{log}_a\frac{M}{N} &= \mathrm{log}_aM - \mathrm{log}_aN \\
\mathrm{log}_aM^r &= r\mathrm{log}_aM
\end{align}
\]
\(\mathrm{log}_aMN = \mathrm{log}_aM + \mathrm{log}_aN\)の証明
\[
\begin{align}
\mathrm{log}_aM &= p \\
\mathrm{log}_aN &= q
\end{align}
\]
とおくと
\[
\begin{align}
M &= a^p \\
N &= a^q
\end{align}
\]
であるから
\[
MN = a^pa^q = a^{p+q}
\]
ゆえに
\[
\begin{align}
左辺 &= \mathrm{log}_aMN = \mathrm{log}_aa^{p+q} = p+q \\
&= \mathrm{log}_aM + \mathrm{log}_aN = 右辺
\end{align}
\]
\[
\huge{Q.E.D.}
\]
\(\mathrm{log}_a\frac{M}{N} = \mathrm{log}_aM - \mathrm{log}_aN\)の証明
\[
\begin{align}
\mathrm{log}_aM &= p \\
\mathrm{log}_a\frac{1}{N} &= q
\end{align}
\]
とおくと
\[
\begin{align}
M &= a^p \\
\frac{1}{N} &= a^q \Leftrightarrow N =a^{-q}
\end{align}
\]
であるから
\[
\frac{M}{N} = \frac{a^p}{a^{-q}} = a^pa^q = a^{p+q}
\]
ゆえに
\[
\begin{align}
左辺 &= \mathrm{log}_a\frac{M}{N} = \mathrm{log}_aa^{p+q} = p+q \\
&= \mathrm{log}_aM + \mathrm{log}_a\frac{1}{N} = \mathrm{log}_aM - \mathrm{log}_a\left\{\frac{1}{N}\right\}^{-1} \\
&= \mathrm{log}_aM - \mathrm{log}_aN = 右辺
\end{align}
\]
\[
\huge{Q.E.D.}
\]
\(\mathrm{log}_aM^r = r\mathrm{log}_aM\)の証明
\[
\mathrm{log}_aM = p
\]
とおくと、
\[
M = a^p
\]
ゆえに
\[
\begin{align}
左辺 &= \mathrm{log}_aM^r = \mathrm{log}_a(a^p)^r = \mathrm{log}_aa^{pr} = pr \\
&= (\mathrm{log}_aM)r = r\mathrm{log}_aM = 右辺
\end{align}
\]
\[
\huge{Q.E.D.}
\]
対数の底の変換公式
\[
\mathrm{log}_ab = \frac{\mathrm{log}_cb}{\mathrm{log}_ca}
\]
\(\mathrm{log}_ab = \frac{\mathrm{log}_cb}{\mathrm{log}_ca}\)の証明
\[
\begin{align}
b &= a^p \\
c &= b^q
\end{align}
\]
とすると、
\[
\begin{align}
p &= \mathrm{log}_ab \\
q &= \mathrm{log}_bc
\end{align}
\]
また、
\[
c = (a^p)^q = a^{pq}
\]
なので、
\[
pq = \mathrm{log}_ac
\]
よって、
\[
pq = (\mathrm{log}_ab)(\mathrm{log}_bc) = \mathrm{log}_ac
\]
\((\mathrm{log}_ab)(\mathrm{log}_bc) = \mathrm{log}_ac\)の両辺を\(\mathrm{log}_bc\)で割ると、
\[
\mathrm{log}_ab = \frac{\mathrm{log}_ac}{\mathrm{log}_bc}
\]
\[
\huge{Q.E.D.}
\]
指数関数のグラフ
対数関数のグラフ
