Given an input, print all exponents where the base and power sum to the input
APL (Dyalog Unicode), 8 5 bytes

⍳*⊢-⍳

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Anonymous prefix tacit function. TIO tests for the range [1..10].
Thanks @lirtosiast for 3 bytes.
How:
⍳*⊢-⍳ ⍝ Tacit function
    ⍳ ⍝ Range. ⍳n generates the vector [1..n].
  ⊢- ⍝ Subtracted from the argument. The vector is now [n-1,n-2,...,0]
⍳*    ⍝ Exponentiate using the range [1..n] as base. The result is the vector
      ⍝ [1^(n-1), 2^(n-2), 3^(n-3),...]