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

6
In practice, a negative logarithm is always expressed so that its
characteristic only is negative, and the negative sign is placed over the
characteristic, which is separated by a point from the decimal part,
which is positive.
When negative logarithms are expressed in this manner, a proper
distinction must be made between the contrary signs of the characteristic and the mantissa, in the operations of their addition, subtraction, and multiplication by any number. Also, in dividing a logarithm
'whose characteristic is negative by any number, the negative characteristic must be made exactly divisible, by adding to it the least negative number which makes it so divisible, and this process must be
corrected by the addition of an equal positive integral number to the
mantissa.
In the system of logarithms whose base is 10, since the logarithms
of all numbers not exact powers of 10, are incommensurable, their
values can be obtained only approximately by decimals.
Hence the logarithms of all numbers greater than 1, not exact
powers of 10, will consist of positive numbers partly integral and
partly decimal, except the logarithms of numbers less than 10: and
the logarithms of all numbers less than 1 will consist of negative:nunmbers partly integral and partly decimal.
As the characteristics of the logarithms of this system can always
be found by inspection, they are omitted, and the mantissc only are:registered in the tables.
9. PROP. To fnd the relation obetween the logarithnms of the same num1er,
bbut of different bases.
Let x, z denote the logarithms of the number z to the bases a, c,:respectively.
Then = a" and x = logaU; u = c, and z = logu.
Whence aT'=', and, taking the logarithms of these equals to
Tbase c.
Then x logca = z logc = z.
but since x = logb and z = logrc.. logu. logca = logu
log-ca
and log-a lo '_
log-a
That is; the logarithm of any number iu to base a, is equal to the
logarithm of the same number u to base e, multiplied by the reciprocal
of the logarithm of a to base c.
This multiplier is called the modgults of the system of logarithms
whose base is a. For the logarithms of all numbers calculated to base
c are converted into logarithms of the same number to base a
by multiplying each logarithm by -oglogi I and.~. logc log, logic 1
Con. If u-c, then loge -, and. log lo g 1.
logv