The term Napierian logarithm or Naperian logarithm, named after the Scottish mathematician John Napier (1550–1617), 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):
NapLog
(
x
)
=
−
10
7
ln
(
x
/
10
7
)
{\displaystyle \operatorname {NapLog} (x)=-10^{7}\ln \left(x/10^{7}\right)}
The Napierian logarithm satisfies identities quite similar to the modern logarithm, such as
NapLog
(
x
y
)
≈
NapLog
(
x
)
+
NapLog
(
y
)
−
161180956
{\displaystyle \operatorname {NapLog} (xy)\approx \operatorname {NapLog} (x)+\operatorname {NapLog} (y)-161180956}
or
NapLog
(
x
y
/
10
7
)
=
NapLog
(
x
)
+
NapLog
(
y
)
{\displaystyle \operatorname {NapLog} \left(xy/10^{7}\right)=\operatorname {NapLog} (x)+\operatorname {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
NapLog
(
θ
)
=
−
10
7
ln
(
sin
(
θ
)
)
{\displaystyle \operatorname {NapLog} (\theta )=-10^{7}\ln(\sin(\theta ))}
Properties
Napier's "logarithm" is related to the natural logarithm by the relation
NapLog
(
x
)
≈
10000000
(
16.11809565
−
ln
x
)
{\displaystyle \operatorname {NapLog} (x)\approx 10000000(16.11809565-\ln x)}
and to the common logarithm by
NapLog
(
x
)
≈
23025851
(
7
−
log
10
x
)
.