Decomposing $\ln(x)$ into sum of even and odd function.

Question

Can somebody help me break $\ln(x)$ into sum of even and odd function. As far as I know every function can be broken in such manner. Not being able to do this as $\ln(-x)$ and $\ln(x)$ cannot exist simultaneously.

Answer 1

The result you're thinking of does not work for $\ln x$ since it takes nonreal values at $x<0$. However, note that $\mathfrak{R}(\ln x)$ is even on $\mathbb{R}\setminus\{0\}$ (where $\mathfrak{R}z$ is the real part of $z$)