Three in ten of the powers of two start with a 1
1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024 — take ten thousand of them
and count only the leading digit. A 1 comes up 3,010 times and a 9 comes
up 458, which is not one ninth of anything. It is not a quirk of two either: the
first digit of bⁿ is decided by the fractional part of n·log₁₀b,
and when that number is irrational the fractional parts spread out evenly and the
digits land in proportion to log₁₀(1 + 1/d). When it is not irrational, they
do not — and the powers of ten start with a 1 every single time.
Count them
Nine digits, and the first one is exactly as likely as the last five put together — the bands for 5 to 9 telescope to log₁₀(10/5), which is log₁₀2, which is the band for 1. The red line is one ninth, which is what you would guess if you had not counted.
Where it comes from
A number starts with the digit d exactly when the fractional part of
its logarithm lands between log₁₀d and log₁₀(d+1). So the
question is only ever: where do those fractional parts land? For 2ⁿ they
are the multiples of 0.30103…, wrapped round — and multiples of an
irrational number, wrapped round, fill the interval evenly.
The two that do not obey, and why not
If the argument is right, it should break in a way you can predict. Powers of ten
have log₁₀10 = 1: the fractional part is zero every time, so it never
leaves the first band. Powers of √10 have a logarithm of exactly one
half, so the fractional parts alternate between two points — and the numbers use two
digits, 1 and 3, and no others ever.
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What this page checks about itself
The powers are computed as exact integers in BigInt, so the leading
digit of 210000 — a number with 3,011 digits — is read off the number
itself and not off a floating-point logarithm.