A minimum requirement: For the simplify function to combine logarithms, they need to be told the arguments are real through assumptions.

Aside from that, you need to make the desired form seem less complex than the default result. That can be achieved through the ComplexityFunction option.

One solution, that penalizes excessive Log[]s:

f[x_] = -(((3 Sqrt[E])/2 - x) (1 - Log[2 Sqrt[E] - x])) + (-(Sqrt[E]/
      2) + x) (1 - Log[x])
dom = FunctionDomain[f[x],x]
g[x_] = PowerExpand /@ (D[f[x], {x, 1}] //
 FullSimplify[#
  , dom
  , ComplexityFunction -> (LeafCount[#] + 5 Count[#, _Log, Infinity]&)
  ]&)

Oddball solution:

f[x_] = -(((3 Sqrt[E])/2 - x) (1 - Log[2 Sqrt[E] - x])) + (-(Sqrt[E]/
      2) + x) (1 - Log[x]);
g[x_] = PowerExpand /@ (D[f[x], {x, 1}] // FullSimplify);
dom = FunctionDomain[f[x], x]
Collect[
 g[x]/.ln_Log:>Hold[1]ln
 ,Hold[1]
 ,Simplify[#, dom, ComplexityFunction -> LeafCount]&
 ] // ReleaseHold

Note: LeafCount gives a slightly different measure of complexity than the default measure Simplify`SimplifyCount:

Simplify`SimplifyCount[-(E/(x*(-2*Sqrt[E] + x)))]
Simplify`SimplifyCount[E/(2*Sqrt[E]*x - x^2)]
(*
Out[122]= 21
Out[123]= 20
*)

LeafCount[-(E/(x*(-2*Sqrt[E] + x)))]
LeafCount[E/(2*Sqrt[E]*x - x^2)]
(*
Out[126]= 17
Out[127]= 18
*)

Here are a couple of ways:

g[x] /. Log[a_] + Log[b_] :> Log[a*b] // Apart
Simplify[g[x], 2 Sqrt[E] - x > 0] // Apart