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)

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