What is "napierian logarithm"

Logarithm \Log"a*rithm\ (l[o^]g"[.a]*r[i^][th]'m), n. [Gr. lo`gos word, account, proportion + 'ariqmo`s number: cf. F. logarithme.] (Math.) One of a class of auxiliary numbers, devised by John Napier, of Merchiston, Scotland (1550-1617), to abridge arithmetical calculations, by the use of addition and subtraction in place of multiplication and division.

Note: The relation of logarithms to common numbers is that of numbers in an arithmetical series to corresponding numbers in a geometrical series, so that sums and differences of the former indicate respectively products and quotients of the latter; thus, 0 1 2 3 4 Indices or logarithms 1 10 100 1000 10,000 Numbers in geometrical progression Hence, the logarithm of any given number is the exponent of a power to which another given invariable number, called the base, must be raised in order to produce that given number. Thus, let 10 be the base, then 2 is the logarithm of 100, because 10^ 2 = 100, and 3 is the logarithm of 1,000, because 10^ 3 = 1,000.

Arithmetical complement of a logarithm, the difference between a logarithm and the number ten.

Binary logarithms. See under Binary.

Common logarithms, or Brigg's logarithms, logarithms of which the base is 10; -- so called from Henry Briggs, who invented them.

Gauss's logarithms, tables of logarithms constructed for facilitating the operation of finding the logarithm of the sum of difference of two quantities from the logarithms of the quantities, one entry of those tables and two additions or subtractions answering the purpose of three entries of the common tables and one addition or subtraction. They were suggested by the celebrated German mathematician Karl Friedrich Gauss (died in 1855), and are of great service in many astronomical computations.

Hyperbolic logarithm or Napierian logarithm or Natural logarithm, a logarithm (devised by John Speidell, 1619) of which the base is e (2.718281828459045...); -- so called from Napier, the inventor of logarithms.

Logistic logarithms or Proportional logarithms, See under Logistic.

The term Napierian logarithm or Naperian logarithm, named after John Napier, is often used to mean the natural logarithm. 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 logarithm):

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

(Since this is 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 logarithm, such as

NapLog(xy) = NapLog(x) + NapLog(y) − 161180950