Replacing $x^x$ by $e^{x\cdot\ln(x)}$

Question

In which case is it allowed to replace a given real valued function such as $f(x)=x^x$ by the term $e^{x\cdot\ln(x)}$?

To be more specific: What properties must a function have so that it's allowed to replace $a^b$ by an exponential function?

Answer 1

If one only considers real valued functions, one has that $$ a^b=e^{b \ln a} $$ holds iff $a>0$. Then one is allowed to write $x^x=e^{x \ln x}$ for any real number $x$ such that $x>0$.

Answer 2

This question is a little bit confusing. $$a^x := \exp\left(x \ln a \right)$$ is the definition of powers with exponents $x\in \mathbf C$ and $a>0$. Therefore expressions like $x^x$ are only for $x>0$ defined.