How to compare sizes/orders using a logarithmic equality?

Questions
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1

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Reduce[Exists[{x, y, z}, {x > 0, y > 0, z > 0, 
     2 + Log[2, x] == 3 + Log[3, y] == 5 + Log[5, z], #}], 
   Reals] & /@ {x > y > z, x > z > y, y > x > z, y > z > x}

Since the solutions to logarithmic equations are not unique, it’s impossible to determine a single exact solution for the equation. The method mentioned above cannot verify the correctness of the third option (Option C). Are there alternative approaches that can perfectly resolve this issue?

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2

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