The natural logarithm of 10 continued fraction is the continued fraction expansion representing 
, beginning [2; 3, 3, 3, 1, 1, 3, 6, 3, 3, 1, 4, 2, 1, 2,
1, ...] (OEIS A016738).
The Engel expansion is 1, 1, 4, 5, 20, 30, 48, ... (OEIS A059182).
The incrementally largest terms in the continued fraction of 
are 2, 3, 6, 26, 716, 774, 982, 1324, 4093, 10322, ...
(OEIS A228346), which occur at positions 0,
1, 7, 17, 30, 136, 962, 1163, 1261, 1293, ... (OEIS A228345).
The plot above shows the positions of the first occurrences of 1, 2, 3, ... in the continued fraction, the first few of which
are 4, 0, 1, 11, 18, 7, 44, 159, 74, 212, 260, 182, 43, 152, 59, 84, 40, 186, 27,
89, ... (OEIS A228270). The smallest positive
integers not occurring in the first 
terms of the continued
fraction are 40230, 45952, 46178, 46530, ... (E. Weisstein, Aug. 18,
2013).
Let the continued fraction of 
be denoted ![[a_0;a_1,a_2,...]](/images/equations/NaturalLogarithmof10ContinuedFraction/Inline5.svg)
and let the denominators
of the convergents be denoted 
, 
, ..., 
. Then plots above show successive values of 
, 
, 
, which appear to converge to Khinchin's
constant (left figure) and 
, which appear to converge to the Lévy
constant (right figure), although neither of these limits
has been rigorously established.
