real slog developed at a fixed point

[Kouznetsov]

...
\( \log(e^z-\el)+r(e^z-\el)=\log(z-\el)+r(z-\el)+\el \)
\( \log\left(\frac{e^z-\el}{z-\el}\right)=r(z-\el)-r(e^z-\el)+\el \)
...

Let
\( f_1=\log(e^z-\el)-\log(z-\el) \)
\( f_2=\log\left(\frac{e^z-\el}{z-\el}\right) \)
Perhaps, it is supposed that \( f_1=f_2 \). I plot both functions below.
   
Levels \( \Re(f)=-2,-1,0,1,2,3,4 \) are shown with thick black curves.
Levels \( \Re(f)=-1.8,-1.6,-1.4,-1.2,-0.8,-0.6,-0.4,-0.2 \) are shown with thin red curves.
Levels \( \Re(f)= 0.2, 0.4, 0.6, 0.8, 1.2, 1.4, 1.6, 1.8 \) are shown with thin red curves.
Levels \( \Im(f)=-2,-1 \) are shown with thick red curves.
Levels \( \Im(f)=1,2 \) are shown with thick blue curves.
Levels \( \Im(f)=\pm \pi,\pm 3\pi \) are shown with thick pink curves.
Levels \( \Im(f)=-4 \) is shown with cyan curve.
Levels \( \Im(f)=-1.8,-1.6,-1.4,-1.2,-0.8,-0.6,-0.4,-0.2 \) are shown with thin red curves.
Levels \( \Im(f)= 0.2, 0.4, 0.6, 0.8, 1.2, 1.4, 1.6, 1.8 \) are shown with thin red curves.