OH GOOD! I’ve been reasoning my way towards this since looking into natural log, wondering “why don’t they use..” and indeed, they do!
“The natural unit of information (symbol: nat), sometimes also nit or nepit, is a unit of information or entropy, based on natural logarithms and powers of e, rather than the powers of 2 and base 2 logarithms, which define the bit. This unit is also known by its unit symbol, the nat. The nat is the coherent unit for information entropy. The International System of Units, by assigning the same units (joule per kelvin) both to heat capacity and to thermodynamic entropy implicitly treats information entropy as a quantity of dimension one, with 1 nat = 1.”