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Jelly, 10 bytes
2*Ḷ^‘l2Ċ⁸_

A dyadic Link that accepts \$w\$ on the left and \$x\$ on the right and yields a list of integers.
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How?
2*Ḷ^‘l2Ċ⁸_ - Link: integer, w; integer x   e.g.  3; 5
2          - two                                 2
 *         - (2) exponentiate (w)                8
  Ḷ        - lowered range                       [0,1,2,3,4,5,6,7] 
   ^       - XOR (x)                             [5,4,7,6,1,0,3,2]
    ‘      - increment                           [6,5,8,7,2,1,4,3]
     l2    - log base two                        [2.58,2.32,3,2.80,1,0,2,1.58]
       Ċ   - ceiling                             [3,3,3,3,1,0,2,2]
        ⁸  - chain's left = w                    3
         _ - subtract                            [0,0,0,0,2,3,1,1]


Without floating points involved, also 10: 2*Ḷ^ḤBẈ⁸_‘
...would be 8 if 0 in binary were [] rather than [0] (2*Ḷ^BẈ⁸_)