关于math:换底公式之推导

\begin{array}{c}换底公式之推导:\\证实:\quad \log_{a}{b}=\frac{\log_{c}{b}}{\log_{c}{a}} \\设:\\\log_{a}{b} =r\\\log_{c}{b} =m\\\log_{c}{a} =n\\即:\\a^r=b\\c^m=b\\c^n=a\\\because a^r=(c^n)^r=b\\\because c^m=b\\\therefore c^m=c^{nr}\\\therefore m=nr\\\because r=\frac{m}{n} \\\therefore \log_{a}{b}=\frac{\log_{c}{b}}{\log_{c}{a}}\\\end{array}