The first definition of the logarithm was constructed by Napier and popularized through his posthumous pamphlet (Napier 1619). It this pamphlet, Napier sought to reduce
the operations of multiplication, division, and root extraction to addition and subtraction.
To this end, he defined the "logarithm" 
of a number 
by
 |
(1)
|
written 
.
This definition leads to the remarkable relations
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
which give the identities
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
(Havil 2003, pp. 8-9). While Napier's definition is different from the modern one (in particular, it decreases with increasing 
, but also fails to satisfy a number of properties of the modern
logarithm), it provides the desired property of transforming multiplication into
addition.
The Napierian logarithm can be given in terms of the modern logarithm by solving equation (1) for 
, giving
 |
(8)
|
Because a ratio of logarithms appears in this expression, any logarithm base 
can be used as long as the same value
of 
is used for both numerator and denominator.
