Using FindInstance we see that all four relationships have exceptions. For condition A:

eqs = {x > 0, y > 0, z > 0, 
   2 + Log[2, x] == 3 + Log[3, y] == 5 + Log[5, z]};
{condsA, condsB, condsC, condsD} = {x > y > z, x > z > y, y > x > z, 
   y > z > x};
FindInstance[Append[eqs, Not[condsA]], {x, y, z}]
Append[eqs, Not[condsA]] /. % // FullSimplify

For condition B:

FindInstance[Append[eqs, Not[condsB]], {x, y, z}]
Append[eqs, Not[condsB]] /. % // FullSimplify

For condition C:

FindInstance[Append[eqs, Not[condsC]], {x, y, z}]
Append[eqs, Not[condsC]] /. % // FullSimplify

For condition D:

FindInstance[Append[eqs, Not[condsD]], {x, y, z}]
Append[eqs, Not[condsD]] /. % // FullSimplify