Proof of the power rule for logarithms What is the proof for the power rule for logarithms? And are there different ways to prove it?

Here is one way to show the identity.  Define the natural logarithm as
$$\log x=\int_1^x \frac{dt}{t}$$
for $x>0$.  Then, we have
$$\log x^n=\int_1^{x^n} \frac{dt}{t}=\sum_{k=1}^n\int_{x^{k-1}}^{x^k}\frac{dt}{t} \tag 1$$
Now substituting $t=x^{k-1}u$ in $(1)$ reveals that
$$\begin{align}
\log x^n&=\sum_{k=1}^n\int_{1}^{x}\frac{du}{u}\\\\
&=n\log x
\end{align}$$
as was  to be shown!
