Napierian logarithm

The term Napierian logarithm or Naperian logarithm, named after John Napier, is often used to mean the natural logarithm. Napier did not introduce this natural logarithmic function, although it is named after him. However, if it is taken to mean the "logarithms" as originally produced by Napier, it is a function given by (in terms of the modern natural logarithm):
 * $$\mathrm{NapLog}(x) = -10^7 \ln (x/10^7) $$

The Napierian logarithm satisfies identities quite similar to the modern logarithm, such as
 * $$\mathrm{NapLog}(xy) \approx \mathrm{NapLog}(x)+\mathrm{NapLog}(y)-161180956$$

or
 * $$\mathrm{NapLog}(xy/10^7) = \mathrm{NapLog}(x)+\mathrm{NapLog}(y) $$

In Napier's 1614 Mirifici Logarithmorum Canonis Descriptio, he provides tables of logarithms of sines for 0 to 90°, where the values given (columns 3 and 5) are


 * $$\mathrm{NapLog}(\theta) = -10^7 \ln (\sin(\theta)) $$

Properties
Napier's "logarithm" is related to the natural logarithm by the relation


 * $$\mathrm{NapLog} (x) \approx 10000000 (16.11809565 - \ln x)$$

and to the common logarithm by


 * $$\mathrm{NapLog} (x) \approx 23025851 (7 - \log_{10} x).$$

Note that


 * $$16.11809565 \approx 7 \ln \left(10\right) $$

and


 * $$23025851 \approx 10^7 \ln (10).$$

Napierian logarithms are essentially natural logarithms with decimal points shifted 7 places rightward and with sign reversed. For instance the logarithmic values


 * $$\ln(.5000000) = -0.6931471806$$
 * $$\ln(.3333333) = -1.0986123887$$

would have the corresponding Napierian logarithms:


 * $$\mathrm{NapLog}(5000000) = 6931472$$
 * $$\mathrm{NapLog}(3333333) = 10986124$$

For further detail, see history of logarithms.