换底公式$\log_{a}{b}=\frac{\log_{c}{b}}{\log_{c}{a}}$的证明
\[proof:\quad \log_{a}{b}=\frac{\log_{c}{b}}{\log_{c}{a}}
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\[设\log_{a}{b} =r,\quad \log_{c}{b} =m,\quad \log_{c}{a} =n
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\[即:a^{r}=b,\quad c^{m}=b,\quad c^{n}=a
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\[\because a^r=(c^n)^r=b
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\[\because c^m=b
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\[\therefore c^m=c^{nr}
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\[\therefore m=nr
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\[\because r=\frac{m}{n}
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\[\therefore \log_{a}{b}=\frac{\log_{c}{b}}{\log_{c}{a}}
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