The decimal expansion of the natural logarithm
of 2 is given by
(OEIS A002162). It was computed to 
decimal digits by S. Kondo on May 14, 2011 (Yee).
The Earls sequence (starting position of 
copies of the digit 
) for 
is given for 
, 2, ... by 4, 419, 2114, 3929, 38451, 716837, 6180096, 10680693,
2539803904 (OEIS A228242).
-constant primes occur at 321, 466, 1271, 15690, 18872,
89973, ... decimal digits (OEIS A228226).
The starting positions of the first occurrence of 
, 1, ... in the decimal expansion of 
are 9, 4, 22, 3, 5, 10, 1, 6, 8, ... (OEIS A100077).
Scanning the decimal expansion of 
until all 
-digit numbers have occurred, the last 1-, 2-, ... digit numbers
appearing are 2, 98, 604, 1155, 46847, 175403, ... (OEIS A036901),
which end at digits 22, 444, 7655, 98370, 1107795, 12983306, ... (OEIS A036905).
The digit string 0123456789 occurs starting at positions 3157027485, 8102152328, ... in the decimal digits of 
, and 9876543210 occurs starting at position 380113805, with
no other occurrences in the first 
digits (E. Weisstein, Aug. 20, 2013).
It is not known if 
is normal, but the following
table giving the counts of digits in the first 
terms shows that the decimal digits are very uniformly
distributed up to at least 
.