\[证明: \quad \log_{n}{a}=\frac{1}{\log_{a}{n}} \]\[\\ \\ \]\[①: \quad \log_{a}{n} = \frac{\lg_{}{n}}{\lg_{}{a}} \]\[\\ \\ \]\[②: \quad \log_{n}{a} = \frac{\lg_{}{a}}{\lg_{}{n}} \]\[\\ \\ \]\[① \Rightarrow \lg_{}{a}=\frac{\lg_{}{n}}{\log_{a}{n}} \]\[\\ \\ \]\[代入②中: \quad \log_{n}{a} = \frac{\frac{\lg_{}{n}}{\log_{a}{n}}}{\lg_{}{n}} \]\[\\ \\ \]\[\Rightarrow \frac{\lg_{}{n}}{\log_{a}{n}} \cdot \frac{1}{\lg_{}{n}}= \frac{1}{\log_{a}{n}} \]\[\\ \\ \]\[\therefore \log_{n}{a}=\frac{1}{ \log_{a}{n} } \] 标签：lg,frac,log,推导,关系式,quad,对数,Rightarrow From： https://www.cnblogs.com/Preparing/p/16755265.html