对数的倒数关系式
\[\begin{align}
推导: \quad \log_{n}{a}=\frac{1}{\log_{a}{n}}
\\ \\
A式: \quad \log_{a}{n} = \frac{\lg_{}{n}}{\lg_{}{a}}
\\ \\
B式: \quad \log_{n}{a} = \frac{\lg_{}{a}}{\lg_{}{n}}
\\ \\
A式 \Rightarrow \lg_{}{a}=\frac{\lg_{}{n}}{\log_{a}{n}}
\\ \\
代入B式中: \quad \log_{n}{a} = \frac{\frac{\lg_{}{n}}{\log_{a}{n}}}{\lg_{}{n}}
\\ \\
\Rightarrow
\frac{\lg_{}{n}}{\log_{a}{n}} \cdot
\frac{1}{\lg_{}{n}}= \frac{1}{\log_{a}{n}}
\\ \\
\therefore \log_{n}{a}=\frac{1}{ \log_{a}{n} }
\\ \\
证明成立
\end{align}
\]
