A peculiar limit? Consider $A^N = X$, where $A$ and $N$ are positive integers and $A > 1$. The ratio between $N$ and the number of digits of $X$ apparently have a limit(!). See these:


*

*$2^3=8$  (ratio $3 : 1$)

*$2^{13}=8192$  (ratio $3.25 : 1$)

*$2^{23}=8388608$  (ratio $3.2857 : 1$ )

*$2^{33}= \ldots$  and so on. 


I used $2$ as a base because no other bases such as $3,4,5,\ldots$ give minimum number of digits for $X$, in other words, $2$ is the only base we need to calculate this limit, And if my calculation is correct, the first $5$ digits of this limit is $3.3219$, Is my calculation correct ?

The limit is,
$$\log_{A}(10)={{\ln(10)} \over {\ln(A)}}$$
Hint:
Prove that the number of digits $D$ is bounded by,
$$\log_{10}(A^N) \le D \le \log_{10}(A^N) +1$$
You'll then want to find,
$$\lim_{N \to {\infty}} {N \over D}$$
