Number of digits - 2

Let $10^{x} = 2^{1000}$ . Then $\log 10^{x} = \log 2^{1000}$ , or $x \log 10 = x = 1000 \log 2$ . Now we make use of the commonly known fact that $\log 2 \approx 0.3010$ . This means $1000 \log 2 \approx 301.0$ . So we know that our number is approximately equal to $10^{301}$ , meaning it has a $1$ followed by $301$ zeroes. Therefore our number of digits is $1 + 301 = \boxed{302}$ .