Problems with ln(ax) equations.

Question

After fiddling around with the ln() function, I arrived at a problem.

I have found that $a \approx 1.39095$. However, I couldn't find the exact value.

Using the Lambert w function, I have already found that for P2 (see image) x=$\frac{1}{w\left ( a \right )}$ And for P3 $$x=\sqrt{\frac{2}{w\left ( \frac{2a^{2}}{e^{2}} \right )}}$$

From there, however, I'm stuck. I can't find the exact solutions for $x\cdot ln\left (a x \right ) - x = ln \left ( ax \right )$, which would help me a great deal. So, how do I continue from here until the end of the question?

Answer 1

Comment on estimates:

As you say, approximations are fairly easy to get, as shown by reg@2:

& fairly easy to get aproximations for any ratio $A:B$ just via trial & error.

Estimates for $A:B=1:1\rightarrow 4$ given by reg@#&/@Range@4:

aa[a_, x_] := Log[a x]
bb[x_] := 1/x
cc[a_, x_] := x Log[a x] - x
reg[n_] := With[{a = {19.6839, 1.39095, 0.729523, 0.507779}[[n]]}, 
With[{p1234 =Join[
{#, aa[a, #]} & /@ {(x /. NSolve[aa[a, x] == cc[a, x], x, Reals])[[1]]}, 
{#, bb@#} & /@ {(x /. NSolve[aa[a, x] == bb[x], x, Reals])[[1]]}, 
{#, bb@#} & /@ {(x /. NSolve[cc[a, x] == bb[x], x, Reals])[[1]]},
{#, aa[a, #]}&/@ {(x /. NSolve[aa[a, x]==cc[a,x],x,Reals])[[2]]}]},
With[{
r1 = ImplicitRegion[cc[a, x] < y < aa[a, x] && y < bb[x], 
{{x, p1234[[1, 1]], p1234[[3, 1]]}, y}],
r2 = ImplicitRegion[bb@x < y < aa[a, x] && y > cc[a, x], 
{{x, p1234[[2, 1]], p1234[[4, 1]]}, y}]},
Column[
{StringJoin["a = ", ToString[a], "; A:B = 1:", 
ToString[Area@r1/Area@r2], "; p1,p2,p3,p4:"], p1234,
  Show[
  Plot[{aa[a, x], bb@x, cc[a, x]}, {x, 0, 2 n}, 
  Epilog -> {Red, PointSize[Medium], Point[p1234]}], 
    RegionPlot[{r1, r2}], ImageSize -> 400]}]]]]