证明对数公式$\log_{a^n}{b^m}=\frac{m}{n} \log_{a}{b}$与$\log_{a}{x^n}=n\log_{a}{x}$
Formula 1
Method 1
\[proof:\quad \log_{a^n}{b^m}=\frac{m}{n} \log_{a}{b}
\]
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\[设\log_{a^n}{b^m}=x
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\[(a^n)^x=b^m \Rightarrow a^{nx}=b^{m}
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\[\log_{a}{b^m}=nx \Rightarrow nx=m\log_{a}{b}
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\[\therefore x=\frac{m\log_{a}{b}}{n} \Rightarrow \frac{m}{n} \log_{a}{b}
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Method 2
\[proof:\quad \log_{a^n}{b^m}=\frac{m}{n} \log_{a}{b} \]
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\[\because \quad \log_{a}{b}=\frac{\log_{c}{b}}{\log_{c}{a}}
\]
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\[\therefore \log_{a^n}{b^m}=\frac{\ln{b^m}}{\ln{a^n}}
=\frac{m\ln{b}}{n\ln{a}} \]
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\[\because \log_{a}{b}=\frac{\ln{b}}{\ln{a}}\]
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\[\therefore \frac{m\ln{b}}{n\ln{a}} =
\frac{m}{n} \cdot \log_{a}{b}
= \log_{a^n}{b^m}
\]
Formula 2
\[proof:\quad \log_{a}{x^n}=n\log_{a}{x}
\]
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\[设\log_{a}{x}=m,\quad 即a^m=x
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\[则\log_{a}{x^n} \Rightarrow \log_{a}{(a^m)^n} \Rightarrow \log_{a}{a^{mn}}
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\[\because a^{m}=x,\quad \log_{a}{x}=m
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\[\therefore \log_{a}{a^m}=m
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\[\therefore \log_{a}{a^{mn}} \Rightarrow mn
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\[\because m=\log_{a}{x}
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\[\log_{a}{a^{mn}} \Rightarrow n \times m \Rightarrow n \times \log_{a}{x}
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\[\therefore \log_{a}{x^n}=n\log_{a}{x}
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