Biggest powers NOT containing all digits.

Let $m>1$ be a natural number with $m \not\equiv 0 \pmod{10}$

Consider the powers $m^n$ , for which there is at least one digit not occurring
in the decimal representation.

Is there a largest $n$ with the desired property for any $m$? If so, define $n(m)$ to be this number.

Examples :

$$m=2 \rightarrow 2^{168} = 374144419156711147060143317175368453031918731001856$$

does not contain the digit $2$.

All the powers above up to $2^{10000}$ conatin all the digits, so
$2^{168}$ seems to be the biggest power with the desired property.

$$m=3 \rightarrow 3^{106} = 375710212613636260325580163599137907799836383538729$$

does not contain the digit $4$.

All the powers above up to $3^{10000}$ contain all the digits, so
$3^{106}$ seems to be the biggest power with the desired property.

So, probably $n(2)=168$ and $n(3)=106$ hold.

Is $n(m)$ defined for any $m$, and if yes, can reasonably sharp bounds be given?

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