Plotted in Maple and converted to SVG format in Inkscape, using the following Maple code:

 
> with(plots); 
> f := proc (n) options operator, arrow; `mod`(n, 10) end proc; 
> g := proc (n) options operator, arrow; `mod`(floor((1/10)*n), 10) end proc; 
> h := proc (n) if 1 <= evalf(log10(n)) then if f(n) = g(n) then return 1+h(floor((1/10)*n)) else return 1 end if else return 1 end if end proc; 
> k := proc (n) if 0 < n then return f(n)+10*h(n)+10^(floor(log10(f(n)+10*h(n)))+1)*k(floor(n/10^h(n))) else return 0 end if end proc; 
> sequence := proc (N) local i; for i to 19 do N[i+1] := [][i][2])] end do end proc; 
> sequencedigits := proc (N) local i; for i to 20 do N[i][2] := floor(log10(N[i][2])) end do end proc; 
> AA := vector(20); AA[1] := [1, 23]; sequence(AA); sequencedigits(AA); a := logplot(AA, color = red); 
> BB := vector(20); BB[1] := [1, 1]; sequence(BB); sequencedigits(BB); b := logplot(BB, color = blue); 
> CC := vector(20); CC[1] := [1, 13]; sequence(CC); sequencedigits(CC); c := logplot(CC, color = violet); 
> DD := vector(20); DD[1] := [1, 312]; sequence(DD); sequencedigits(DD); d := logplot(DD, color = green); 
> display(a, b, c, d, view = [1 .. 20, 1 .. 10000]);