What is the time complexity of finding the most significant digit of $3\uparrow\uparrow n$?

Question

What is the time complexity of finding the most significant digit of $3\uparrow\uparrow n$? I know we can find the least significant digits in constant time using modular arithmetic, is the most significant digit known to be harder to compute?

Answer 1

The first digit of $$3\uparrow\uparrow 4$$ still can be determined. We have $$\log_{10}(3)\cdot 3^{(3^3)}=3638334640024.099685574579370$$

Since $10^{0.1}<2$ , we can conclude that the first digit is $1$.

But to calculate the first digit of $$3\uparrow\uparrow 5$$ , we need the logarithm of $3\uparrow\uparrow 5$ with precision better than $1.0$

With a supercomputer, it might be possible to calculate the logarithm precise enough. The magnitude of the logarithm is roughly $10^{(10^{12.56)}}$.

The first digit of $3\uparrow\uparrow n$ for any $n\ge 6$ cannot be caclaulated because the logarithm simply gets too large.