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

2
A system of logarithms may be calculated to any base except
unity, and lence there may be an indefinite number of systems of
logarithms, according to the different assumptions made for the bases.
There are, however, only two systems of logarithms used by mathematicians, one for shortening numerical calculations and the. other in
analytical reasonings.
The following consequences may be shown to arise from the eqiation ut = ga10TU
The logarithm of 1 is 0, or logal = 0; and the logarithm of the
base is 1, or loga = 1.
If logat be positive, and assume successively and continuously all
possible values from 0 to + co, it is obvious that it will receive all
values from 1 to oo.
If logan be negative, and assume all possible values front
0 to- oo, u will receive all values from 1 to 0.
Hence, as logdit changes continuously from + 0 to - o, -
changes continuously from +0 to 0, and consequently produces all
the positive natural numbers.
If the base a be 10 and remain constant, and it be madle to
assume successively 1, 2, 3, 4, &c., the corresponding values of x in
the equation it 10I, when comlputed and registered will form a table
of that system of logarithmls whose base is 10.
2. PRoP. To find thle logarith/ie of tAe product of two 2number'S
HIere i, = aOlga", and it2 = aClogt2, by lef.
it..?I t2 = aloSatio. (Cloa lat2 = (tlo tul +logaUt2
And loga {u. iG} =logai + logan, by def.
Or, the logarithln of the product of two numbers, is equal to the
sull of the logarithms of the numbers themselves.
Con. In a similar way it may be shewn that the
loga{uI2. 2t3 ~;. ~ } -logul1 + loga'2 + logq -?.. +.
Or that the logarithm of the product of any number of factors, is
equial to the sum of the logarithms of the several factors.
3. PROP. To find the logarithti of the quotient of two nu21trs.
Here i 1= al~lOg,, and tz.,a —= aog~0a2 by def.
Then i = ao10al - 0loga^
~l2 alo10^,,l
** 10 l{oga - } g log10,,1
Or, the logarithm of a quotient, is equal to the difference arising
fiom subtracting the logarithm of the divisor from the logarithm of
the dividend.
Conrt. log, { -- = logai- loga2 = lo - {log.-logl} = -log,. U"~2)~C na Ij j