Elementary arithmetic, with brief notices of its history... by Robert Potts.

THE PROPERTIES AND CONSTRUCTION OF
LOGARITHMS.
ART. 1. DEE. 1. In the equation u = a', where a is a constant number
ireater than unity, and u any natural number, the index x is defined
to be the logarithm of the number u to the base a.
The notation assumed to denote "the logarithm of the number X
to the base a" is logau, so that x=log,,s, and the equation ut-a
may be written u = agau.
DEF. 2. The base of any system of logarithms is any fixed number
which, being raised to the powers denoted by the logarithms, produces the successive natural numbers.
DEF. 3. A system of logarithms is a series of the successive values
of x derived from the equation t=a', when the natural numbers
0, 1, 2, 3, 4, &c., are successively substituted for uz, the same base a
being preserved.1
Logarithms may be defined to be, as in fact they are, a series of numbers in
Arithmetical progression which increase by a common difference, corresponding to
nnother series in Geometrical progression which increase by a common multiplicr.
For example, let 10 be made the base,
If 0, 1, 2, 3, 4, 5,.. be a series in A. P.
and 100, 10, 1, 103, 10, 100,... 10'I 1 be a corresponding series in
or 1, 10, 100, 1000, 10000, 100000... J G. P.
Then the numbers 0, 1, 2, 3, 4, &c., are the logarithms of the series of numbers
1, 10, 100, 1000, 10000 &c., respectively, to the base 10.
Hence it is obvious that a negative number cannot be assumed as the base of any
system of logarithms; for the odd powers of a negative number are negative, andt
the even powers are positive, and consequently they are not subject to the law of
coritiiiuity in producing in order all the natural numbers.
This definition of a system of logarithms suggests a method of finding the
logarithms of all the intermediate numbers; for the Arithmetic mean between any
two consecutive terms of the Arithmetic series will be the logarithm of the
Geometric mean of the two corresponding terms of the Geometric series:
Thus, the A. mean between 0 and 1 is '5,
and the G. mean between 1 and 10 is 3'1622777;
Hence '5 is the logarithm of 3'1622777.
Again the A. mean between 1 and 2 is 1 '5,
and the G. mean between 10 and 100 is 31-6227766;
Hence 1'5 is the logarithm of 31'6227766;
and so on for successive mean proportionals.
Next, the A. means can be found between every two consecutive terms of the
A. series 0, '5, 1, 1'5, &c., and the G. means between every two corresponding terms
of the G. series 1, 3-1622777, 10, 31-6227766, 100, &c.; and so continuing the same
process, may be found the logarithms of all numbers, but at very great expense of
time and labour.