Why is $\ln(x^x)=x\ln(x)$ valid? I know that $\ln(x^k)=k\ln(x)$ for any constant $k$, but why is $\ln(x^x)=x\ln(x)$. The exponent $x$ is not constant.

As $x$ is probably not an integer, $x^x$ is defined as :
$$x^x = e^{x\ln(x)}$$
Hence, taking the logarithm give you $\ln{x^x}=x\ln(x)$
