The logarithmic density of a set 
of positive integers, when the limit exists, is
 |
Logarithmic density weights smaller integers more heavily than natural density. If a set has natural density 
, then it also has logarithmic density
,
but the converse need not hold.
This density occurs naturally in results about multiplicative number theory and iterated arithmetic maps. For example, Tao (2022) formulated his almost-all result for the Collatz conjecture using logarithmic density.
