Smallest Diversifying Exponent
Brachylog (v2), 9 bytes

;.≜^dl10∧


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Explanation

;.≜^dl10∧
 .≜        Brute-force all integers, outputting the closest to 0
;  ^         for which {the input} to the power of the number
    d        has a list of unique digits
     l10     of length 10
        ∧  (turn off an unwanted implicit constraint)