How to ensure that the axes of both plots are of identical length?

Question

I currently have two plots in tikz. There goes the MWE:
documentclass[10pt]{standalone}

% PACKAGES LOADING

usepackage{tikz} % To plot almost everything.
usepackage{pst-3d, tikz-3dplot} % To draw in 3D.

% FIGURES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

begin{document}

begin{tikzpicture}[my plot/.style={thick, smooth, samples=100, domain=0.1:5}, my grid/.style={densely dotted,opacity=0.45, every node/.style={black,opacity=1}}, my axis/.style={latex-latex}]
draw[my plot] (0,0) plot (x,{ln(x)});
coordinate (start plot) at (0.1,{ln(0.1)});
coordinate (end plot) at (5,{ln(5)});
draw[my axis] ([shift={(-0.5cm,0.5cm)}]start plot |- end plot) node[above] {$u(cdot)$} |- node[coordinate](origin){} ([shift={(0.5cm,-0.5cm)}]start plot -| end plot) node[right] {$cdot$};
defx{0.5}defy{4}defp{0.55}
coordinate (Ux) at (x,{ln(x)});
coordinate (Uy) at (y,{ln(y)});
coordinate (Up) at ({p*x+(1-p)*y},{ln(p*x+(1-p)*y)});
draw (Ux) -- coordinate[pos=1-p] (Up-mid) (Uy);
path let p1=(Up-mid), n1={pow(e,y1*0.03514)} in (28.4576*n1,y1) coordinate (Up-mid2);
draw[my grid] (Ux) |- node[below]{$x$} (origin) |- node[left]{$u(x)$} cycle;
draw[my grid] (Uy) |- node[below]{$y$} (origin) |- node[left]{$u(y)$} cycle;
draw[my grid] (Up) |- node[below, yshift=1.4pt]{$px+(1-p)y$} (origin) |- node[left]{$u(px+(1-p)y)$} cycle;
draw[my grid] (Up-mid) |- (origin) |- node[left]{$pu(x)+(1-p)u(y)$} cycle;
draw[my grid] (Up-mid) -- (Up-mid2);
end{tikzpicture}

begin{tikzpicture}[my plot/.style={thick, smooth, samples=100, domain=0.1:5.5}, my grid/.style={densely dotted,opacity=0.5, every node/.style={black,opacity=1}}, my axis/.style={latex-latex}]
draw[my plot] (0,0) plot (x,{(x)});
coordinate (start plot) at (0,{(0)});
coordinate (end plot) at (5.5,{(5.5)});
draw[my axis] ([shift={(-0.5cm,0.5cm)}]start plot |- end plot) node[above] {$u(cdot)$} |- node[coordinate](origin){} ([shift={(0.5cm,-0.5cm)}]start plot -| end plot) node[right] {$cdot$};
defx{0.5}defy{4}defp{0.55}
coordinate (Ux) at (x,{(x)});
coordinate (Uy) at (y,{(y)});
coordinate (Up) at ({p*x+(1-p)*y},{(p*x+(1-p)*y)});
draw (Ux) -- coordinate[pos=1-p] (Up-mid) (Uy);
path let p1=(Up-mid), n1={pow(e,y1*0.03514)} in (28.4576*n1,y1) coordinate (Up-mid2);
draw[my grid] (Ux) |- node[below,]{$x$} (origin) |- node[left]{$u(x)$} cycle;
draw[my grid] (Uy) |- node[below]{$y$} (origin) |- node[left]{$u(y)$} cycle;
draw[my grid] (Up) |- node[below, yshift=1.4pt] {$px+(1-p)y$} (origin) |- node[align=right,xshift=-55.0pt]{$pu(x)+(1-p)u(y)=$$=u(px+(1-p)y)$} cycle;
end{tikzpicture}

end{document}

As it can be seen in the output, both plots have axes of different length. What can I do to make sure that the y-axis and x-axis have the same length in both plots?
Thank you all for your time.

Zarko · Accepted Answer

With pgfplots is not so hard:
documentclass{article}
usepackage{pgfplots} 
pgfplotsset{compat=1.17}
usetikzlibrary{arrows.meta,
                intersections,
                positioning}
pgfplotsset{width=6cm, height=6cm, % <---
    axis lines=middle,
    xlabel={$cdot$},
    xlabel style={anchor=west},
    ylabel={$u(cdot)$},
    ylabel style={anchor=south},
    xtick=empty,   ytick=empty,
    clip=false
            }
tikzset{
ddline/.append style = {draw=gray, densely dotted},
         tick/.style = {inner sep=2pt, font=tiny, align=right}
        }

begin{document}
    begin{figure}[ht]
    begin{tikzpicture}[baseline]
begin{axis}[
       xmin = 0,
     domain = 0.1:5,
    samples = 100
            ]
addplot [thick] {ln(x)};
%
draw[ddline]   (0.5,0) node[tick,above] {$x$} |- coordinate[pos=0.5] (ux)
                (0,{ln(0.5)})   node[tick,left] {$u(x)$};
draw[ddline, name path=B]
    (2,0) node[tick,below]    {$px+(1-p)y$} |-
    (0,{ln(2)})   node[tick,left]     {$p(u)x+(1-p)u(y)$};
draw[ddline]   (4,0) node[tick,below]    {$y$} |- coordinate[pos=0.5] (uy)
                (0,{ln(4)})   node[tick,left]     {$u(y)$};
draw[name path=A, semitransparent]   (ux) -- (uy);
draw[name intersections={of=A and B, by=s}, ddline]
    (s) -- (s -| 0,0) node[tick,left] {$u(px+(1-p)y)$};
end{axis}
    end{tikzpicture}%
hfill
    begin{tikzpicture}[baseline]
begin{axis}[
    xmin=0, ymin=0
            ]
addplot [thick] coordinates {(0.1,0.1) (5,5)};
%
draw[ddline]
    (0.5,0) node[tick,below] {$x$}          |- (0,0.5) node[tick,left] {$u(x)$}
    (2.0,0) node[tick,below] {$px+(1-p)y$}  |- (0,2.0) node[tick,left] {$p(u)x+(1-p)u(y)=$
                                                                        $=u(px+(1-p)y)$}
    (4.0,0) node[tick,below] {$y$}          |- (0,4.0) node[tick,left] {$u(y)$};
end{axis}
    end{tikzpicture}
    end{figure}

or

begin{figure}[ht]
    begin{tikzpicture}[baseline]
begin{axis}[
    axis x line shift={-ln(0.1)}, % <---
       xmin = 0,
     domain = 0.1:5,
    samples = 100
            ]
addplot [thick] {ln(x)};
%
draw[ddline]   
    (0.5,{ln(0.1)}) node[tick,above] {$x$} |- coordinate[pos=0.5] (ux)
    (0,{ln(0.5)})   node[tick,left] {$u(x)$};
draw[ddline, name path=B]
    (2,{ln(0.1)}) node[tick,below]    {$px+(1-p)y$} |-
    (0,{ln(2)})   node[tick,left]     {$p(u)x+(1-p)u(y)$};
draw[ddline]   
    (4,{ln(0.1)}) node[tick,below]    {$y$} |- coordinate[pos=0.5] (uy)
    (0,{ln(4)})   node[tick,left]     {$u(y)$};
draw[name path=A, semitransparent]   (ux) -- (uy);
draw[name intersections={of=A and B, by=s}, ddline]
    (s) -- (s -| 0,0) node[tick,left] {$u(px+(1-p)y)$};
end{axis}
    end{tikzpicture}%
hfill
    begin{tikzpicture}[baseline]
begin{axis}[
    xmin=0, ymin=0
            ]
addplot [thick] coordinates {(0.1,0.1) (5,5)};
%
draw[ddline]
    (0.5,0) node[tick,below] {$x$}          |- (0,0.5) node[tick,left] {$u(x)$}
    (2.0,0) node[tick,below] {$px+(1-p)y$}  |- (0,2.0) node[tick,left] {$p(u)x+(1-p)u(y)=$
                                                                        $=u(px+(1-p)y)$}
    (4.0,0) node[tick,below] {$y$}          |- (0,4.0) node[tick,left] {$u(y)$};
end{axis}
    end{tikzpicture}
    end{figure}
end{document}

How to ensure that the axes of both plots are of identical length?

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