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The Napierian logarithm, as first defined by John Napier, is a function which can be defined in terms of the modern logarithm by:

${\displaystyle \mathrm {NapLog} (x)={\frac {\log {\frac {10^{7}}{x}}}{\log {\frac {10^{7}}{10^{7}-1}}}}.}$

(Being 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: ${\displaystyle \mathrm {NapLog} (x)=\log _{\frac {10^{7}}{10^{7}-1}}10^{7}-\log _{\frac {10^{7}}{10^{7}-1}}x}$ and hence it is a linear function of a particular logarithm, and so satisfies identities quite similar to the modern one.

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