밑변환 change of base

로그 밑변환 (change of base for logarithms)

\(
\log_{a}{b}=x \\
\Rightarrow a^x=b \\
\Rightarrow \log_{c}(a^x)=\log_{c}{b} \\
\Rightarrow x\log_{c}{a}=\log_{c}{b} \\
\Rightarrow x=\frac{\log_{c}{b}}{\log_{c}{a}}
\Rightarrow \log_{a}{b}=\frac{\log_{c}{b}}{\log_{c}{a}}
\)

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지수 밑변환 (change of base for exponents)

\(
a^{\log_{c}{b}}=x \\
\Rightarrow \log_{c}(a^{\log_{c}{b}})=\log_{c}x \\
\Rightarrow \log_{c}{b}\times\log_{c}a=\log_{c}x \\
\Rightarrow \log_{c}(b^{\log_{c}{a}})=\log_{c}x \\
\Rightarrow b^{\log_{c}{a}}=x \\
\Rightarrow b^{\log_{c}{a}}=a^{\log_{c}{b}}
\)