I have raised https://github.com/elastic/elasticsearch/issues/13511 to update the documentation but in the mean time, from looking at the code in FieldValueFactorFunction.Modifier:
for a value n:
- log1p = Math.log10(n + 1)
- log2p = Math.log10(n + 2)
- ln1p = Math.log1p(n) = Math.log(n + 1) [= ln(n + 1)]
- ln2p = Math.log2p(n + 1) = Math.log(n + 2) [= ln(n + 2)]
Hope that helps