Napierian logarithm

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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:

A plot of the Napierian logarithm for values between 0 and 108.
A plot of the Napierian logarithm for values between 0 and 108.

\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:

\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

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

and to the common logarithm by

\mathrm{NapLog} (x) \approx 23025850 (7 - \log_{10}(x)).


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