Powers of special class of positive integers whose representation in a base consists of digits only powers of that integer

Question

For an integer $m \in (\sqrt {10} , 10)$ , define $A_{10,m}:=\{n \in \mathbb N : m^n=\sum_{j=0}^k 10^j m^{n_j} ; n_j=0 $ or $1; k \ge 0\}$ . So , $A_{10,m}$ is the set of those natural numbers , raised to which power of $m$ , its digits in base $10$ representation also consists of powers of $m$ only . Then can we say that for which such integers $m \in (\sqrt {10} , 10)$ , is $A_{10,m}$ a finite set ? In general what can we say if $10$ is replaced by some $d \ge 3$ ?