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Napierian logarithm

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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:

A plot of the Napierian logarithm for values between 0 and 10.

N a p L o g ( x ) = log ⁡ 10 7 x log ⁡ 10 7 10 7 − 1 . {\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:

N a p L o g ( x ) = log 10 7 10 7 − 1 ⁡ 10 7 − log 10 7 10 7 − 1 ⁡ x {\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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