How many digits does 2 "raise to" 1000 have?

We have to find decimal (base 10) digits. So convert to base 10.

2¹⁰⁰⁰ = 10^{(log 2)1000}

Now 10¹, 10², and 10³ has 2, 3, and 4 digits respectively.

So 2¹⁰⁰⁰ should have approximately (log 2)1000 + 1 digits.