2- Number of Digits

The number of digits of any number $N$ is given by

$\lceil \log_{10} N \rceil$

Applying the formula here,

$\lceil \log_{10} 2^{128} \rceil = \lceil 128 \log_{10} 2 \rceil = \lceil 128 \times 0.3010\rceil = \boxed{39}$