What is the number of digits of this number: $2^{333111160}$?

Question

My question is: What is the number of digits of this number? : $$2^{333111160}$$

Answer 1

$$ 333111160 = \log_2 2^{333111160} = \frac{\log_{10}333111160}{\log_{10}2} $$ $$ \text{So }\log_{10} 333111160 = 333111160\log_{10}2 \approx 100276451.05\ldots $$ This is between $100276451$ and $100276452$, so the number has $100276452$ digits.

But this answer is only as good as the calculator I used.

Answer 2

$$2^{333111160} \sim 10^{(\log_{10} 2) 333111160} \sim 100266459\text{ digits}$$