How to calculate number of digits of a large number?

Question

Does anyone know any efficient ways of finding the number of digits in the large number $N = 4^{4^{4^4}}$? Thanks.

Answer 1

This is a really interesting topic! I googled around a bit and discovered a question which somebody else asked, which looks similar to yours. And below it is an excellent answer. Finding the number of digits of a large number.

I hope this helps.

Seraphina

Answer 2

An approximation: $4^{4^4} = 4^{256}$ is approximately $1.34078079 \times 10^{154}$

So the number of digits in $4^{4^{4^4}}$ is approximately $\log_{10}4\times1.34078079 \times 10^{154}$ which is about $8.0723047\times10^{153}$.

It would not be too arduous to (get a computer to) perform this calculation exactly.

Update
The number of digits required is $\lfloor 4^{256}\times \log_{10}4 \rfloor + 1 = \lfloor 2^{513}\times \log_{10}2 \rfloor + 1$. If $\log_{10}2$ is calculated in binary, the multiplication by $2^{513}$ is just a matter of shifting bits, and the problem is reduced to calculating $\log_{10}2$ with the necessary accuracy, which admittedly is not simple.
For completeness, here is the entire WolframAlpha calculation.