The Wikipedia page for 1024 gives a handy technique for estimating large powers of 2 in decimal notation. For exponents up to about 100,

2^10a+b ≈ 2^b10^3a.

For example, 2³⁵ = 34359738368 ≈ 32 × 10⁹ = 32000000000.

This works because 2¹⁰ ≈ 10³. 3a gives a good estimate of the number of digits for exponents up to 300.

(Thanks, Stephen.)