Napierian logarithm

$mathrm\{NapLog\}(x)\; =\; frac\{log\; frac\{10^7\}\{x\}\}\{log\; frac\{10^7\}\{10^7\; -\; 1\}\}.$

(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

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