换底公式之推导

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\begin{array}{c}
换底公式之推导:\\
证明:\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}