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The term Napierian logarithm, or Naperian logarithm, is often used to mean the ], but as first defined by ], it is a function which can be defined in terms of the modern ] by: The term '''Napierian logarithm''', or Naperian logarithm, is often used to mean the ], but as first defined by ], it is a function which can be defined in terms of the modern ] by:


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<math>\mathrm{NapLog} (x) \approx 23025850 (7 - \log_{10}(x))</math>. <math>\mathrm{NapLog} (x) \approx 23025850 (7 - \log_{10}(x))</math>.


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Revision as of 09:10, 30 September 2008

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 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.

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

N a p L o g ( x ) ≈ 9999999.5 ( 16.11809565 − ln ⁡ ( x ) ) {\displaystyle \mathrm {NapLog} (x)\approx 9999999.5(16.11809565-\ln(x))}

and to the common logarithm by

N a p L o g ( x ) ≈ 23025850 ( 7 − log 10 ⁡ ( x ) ) {\displaystyle \mathrm {NapLog} (x)\approx 23025850(7-\log _{10}(x))} .

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