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The term Napierian logarithm, or Naperian logarithm, is often used to mean the natural logarithm, but as first defined by John Napier, it 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.

The Napierian logarithm is related to the natural logarithm by the relation

${\displaystyle \mathrm {NapLog} (x)\approx 9999999.5(16.11809565-\ln(x))}$

and to the common logarithm by

${\displaystyle \mathrm {NapLog} (x)\approx 23025850(7-\log _{10}(x))}$.

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