If we listed the first digits of each of the first 100 powers of 2, we would get:
Looking at these digits, not each digit appears with equal frequency. For example, the digit 1 appears 30 times (or 30% of the time), while the digit 9 only appears 5 times.
If you think about it, this makes sense. If you take any number, the first digit of its double will be 1 if the first digit of the original number is anywhere between 5 and 9, while the first digit of its double will be 9 only if the first two digits of the number are between 45 and 49.
If we take a look at the first digits of the first few numbers of the Fibonacci sequence, the results are similar:
Looking at the first 30 numbers, 1 appears 9 times, or, again, 30% of the time, while 9 only appears once.
Benford's law, published by Frank Benford in 1938, states that, in various lists of numbers, the digit 1 appears in the leftmost position about 30% of the time, much greater than the 11.1% that would result if each digit occurred with equal probability. In general, in a number list, the probability of a digit appearing in the leftmost position is equal to log(1 + 1/n), where n is the digit. This gives the following probabilities, approximately: