The term Napierian logarithm, or Naperian logarithm, is often used to mean the natural logarithm. However, as first defined by John Napier, it is a function given by (in terms of the modern logarithm):
(Since this is a quotient of logarithms, the base of the logarithm chosen is irrelevant.)
It is not a logarithm to any particular base in the modern sense of the term; however, it can be rewritten as:
and hence it is a linear function of a particular logarithm, and so satisfies identities quite similar to the modern one.
The Napierian logarithm is related to the natural logarithm by the relation
and to the common logarithm by