## Calculus with Applications (10th Edition)

$\log_aX^r=\log_a(X.X.X...X)$ with $X.X.X...X$ equals to $X^r$ We have the property $\log_axy=\log_ax+\log_ay$ so $\log_a(X.X.X...X)=\log_aX+\log_aX+...+\log_aX$ with $\log_aX+\log_aX+...+\log_aX$ is $r$ times of $\log_aX$ $\rightarrow \log_aX+\log_aX+...+\log_aX=r\log_aX$ Hence, $\log_aX^r=r\log_aX$