The domain of  $f\left(x\right) \mathrm{is} \left(0,1\right)$ Then the domain of $f\left(e^x\right)+f\left(\ln \left|x\right|\right)$ is

$\begin{array}{l}f\left(x\right) \mathrm{is} \mathrm{defined} \mathrm{for} x\in \left(0,1\right)\\ \mathrm{So}, f\left(e^x\right)+f\left(\ln \left|x\right|\right) \mathrm{is} \mathrm{deffined} \mathrm{for}\\ 0<e^x<1 \mathrm{and} 0<\ln \left|x\right|<1\end{array}$

$\begin{array}{l}\mathrm{or} -\infty <x<0 \mathrm{and} 1<\left|x\right|<e\\ \mathrm{or} x\in \left(-\infty ,0\right) \mathrm{and} x\in \left(-e,-1\right)\cup \left(1,e\right)\\ \mathrm{or} x\in \left(-e,-1\right)\end{array}$