Computing the last digits of the powers of $n$ can be done efficiently with modular exponents. However computing the first digits of the powers of $n$ efficiently is not trivial for base 10.
And so one might naturally begin looking for patterns in the powers of $n$. We generate the first digit of $2^n$ up to a large $n$, then identify sequences which seemed to repeat often & highlight them. After doing so, the number of columns is adjusted to find a pattern. Using a columns of 10 digits, and highlighting some particular sequences the following pattern emerges: (link for for more):
What is the pattern?
For each first digit of a power of 2, there are only one or two possible first digits of the next power of 2 (for instance: doubling a number with first digit 3 can only give a number with first digit 6 or 7; doubling a number with first digit 7 can only give a number with first digit 1). Thus the first digits can only appear in the cycles 1>2>5>1; 1>3>6>1; 1>3>7>1; 1>2>4>9>1; 1>2>4>8>1. In general, the sequence will change course from one of these to any other at a first digit of 1, but unpredictably, in the long run, because as the comments point out, $\log_{10}2$ is irrational. The reason why columns of 10 creates a considerable degree of regularity is just that $2^{10} \approx 10^3$.