\[proof:\quad \log_{a}{b}=\frac{\log_{c}{b}}{\log_{c}{a}} \]\[\\ \\ \]\[设\log_{a}{b} =r,\quad \log_{c}{b} =m,\quad \log_{c}{a} =n \]\[\\ \\ \]\[即:a^{r}=b,\quad c^{m}=b,\quad c^{n}=a \]\[\\ \\ \]\[\because a^r=(c^n)^r=b \]\[\\ \\ \]\[\because c^m=b \]\[\\ \\ \]\[\therefore c^m=c^{nr} \]\[\\ \\ \]\[\therefore m=nr \]\[\\ \\ \]\[\because r=\frac{m}{n} \]\[\\ \\ \]\[\therefore \log_{a}{b}=\frac{\log_{c}{b}}{\log_{c}{a}} \] 标签：because,frac,log,换底,therefore,quad From： https://www.cnblogs.com/Preparing/p/16586228.html