Jelly, 12 11 bytes
1*@ṾØDfƑʋ1#
How it works
1*@ṾØDfƑʋ1# Main link. Argument: n
1 Set the return value to 1.
1# Call the link to the left for k = 1, 2, ... and with right argument n,
until it returns a truthy value.
ʋ Combine the four links to the left into a dyadic chain.
*@ Compute n**k.
Ṿ Convert it to its string representation.
ØD Yield "0123456789".
fƑ Filter and return 1 is the result is equal to the left argument.