Napierian Logarithm

$\begin{figure}\begin{center}\BoxedEPSF{NapierianLogarithm.epsf}\end{center}\end{figure}$

Write a number as

$\begin{displaymath} N=10^7(1-10^{-7})^L, \end{displaymath}$

then

is the Napierian logarithm of

. This was the original definition of a Logarithm, and can be given in terms of the modern Logarithm as

$\begin{displaymath} L(N)=-{\log\left({n\over 10^7}\right)\over\log\left({10^7\over 10^7-1}\right)}. \end{displaymath}$

The Napierian logarithm decreases with increasing numbers and does not satisfy many of the fundamental properties of the modern Logarithm, e.g.,

$\begin{displaymath} \mathop{\rm Nlog}\nolimits(xy)\not=\mathop{\rm Nlog}\nolimits x+\mathop{\rm Nlog}\nolimits y. \end{displaymath}$