Drawing the graph of the logarithm of a function

Question

I am trying to plot a function, which involves taking the logarithm of another function, over the domain (0,1).

The graph is not smooth as it nears 0, as you can see in the image below, which it produces. In fact, it is not well-behaved quite a long way away from 0. But it appears well around 1.

How can I produce a graph that appears smooth over the whole domain?

Code v1

\documentclass{standalone}
\usepackage{tikz,pgfplots}
\begin{document}
\begin{tikzpicture}
\begin{axis}
    [xmin=0,xmax=1,xtick={0,1},ymin=0,ymax=0.12,ytick={0}]
    \addplot[domain=0:1,samples=100]{(1/2)*((1 - x)/x)*((ln((1 + x)/(1-x))/x) - 2)};
\end{axis}
\end{tikzpicture}
\end{document}

Output v1

UPDATE:

Rearranging the expression to make the arguments in ln() simpler seems to help ln((1+x)/(1-x)) = ln(1+x)-ln(1-x). I also specified \pgfplotsset{compat=1.18} as specified by KersouMan, but that did not help me much, as you can see. I also added restrict y to domain={0:0.12} to get rid of some big spikes. Finally, I thought I would give it a break and remove values close to 0 from the domain, instead using domain=0:1, but I do not want to do that, really I want pgfplots to graph this smoothly to (0,0).

Any more ideas?

Code v2:

\documentclass{standalone}
\usepackage{tikz,pgfplots}
\pgfplotsset{compat=1.18}
\begin{document}
\begin{tikzpicture}
\begin{axis}
    [xmin=0,xmax=1,xtick={0,1},ymin=0,ymax=0.12,ytick={0}]
    \addplot[domain=0.03:1,samples=100,restrict y to domain={0:0.12}]{(1/2)*((1 - x)/x)*( (ln(1 + x) - ln(1 - x))/x - 2)};
\end{axis}
\end{tikzpicture}
\end{document}

Output v2

Specific replies:

Re: KersouMan - Unfortunately, that did not help me, I wonder how it helped you.

Re: Jasper Habicht - It is a smooth function that rises from y=0 when x=0 and tends back to y=0 when x=1. Yes, I also believe the issue is that pgfplots is having issues near 0 due to the function getting complicated.

Answer 1

A workaround: use the Taylor polynomial for the logarithmic part:

\documentclass[border=1.618]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.18}

\begin{document}
\begin{tikzpicture}
\begin{axis}[xmin=0,xmax=1,xtick={0,1},ymin=0,ymax=0.12,ytick={0}]
  \addplot[domain=0.4:1,samples=100] {(1/2)*((1 - x)/x)*( (ln(1 + x) - ln(1 - x))/x - 2)};
  \addplot[red,domain=0.0001:0.4,samples=41]{(1 - x)*(x/3 + x^3/5 + x^5/7 + x^7/9)};
\end{axis}
\end{tikzpicture}
\end{document}