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

3

Or, the logarithm of any fraction is equal to the logarithmn of its
reciprocal taken negatively.1
4. PuOP. To find the logaritlmn of any power of a number.
Hlere u = aloga0 by clef.
Then raising each to the 2ih power.
-a = alloa11&..l. log,,{{I} log}n.
Or, the logarithm of any power of a number, is equal to thle product
of the logarithm of the number and the index of the power.
5. PrOP. To find the logaritlhm of any root of a number.
Here = Cal0clt by def.
t 1 1
Then 1' = C a
And loga{ U6^} = -10og6.
Or, the logarithm of any root of a number, is equal to the quotient
arising from dividing the logarithm of the number by the index of
the root.
Hence it appears that if a table of the logarithms of the natural
numbers be arranged in order; by means of them can be performed
the operations of multiplication, division, involution and evolution of
all numbers within the limits of the table. Thus, if one number is to
be multiplied or divided by another, by taking their logarithms from
tile table, and adding or subtracting them, and then by finding'in thetable the number whose logarithm is equal to the sum or difference,
the product or quotient of the two numbers is found. And the power
or root of a number is found by taking the logarithm of the number
from the table, and multiplying or dividing it by the index of the
power or root, and then by finding in the table the number whose
logarithm is equal to this product or quotient, the power or root of the
proposed number is determined. Thus, by the aid of a table of
logarithms, the arithmetical operations of multiplication and division
may be effected by addition and subtraction: and those of involution
or evolution by multiplying or dividing the logarithm by the index
of the power or root. These are the advantages of logarithms in
effecting numerical computations.
I Care must be taken not to confound the expressions loga I- and log, q.: the
fonlrer being the logarithm of the quotient of two numbers, which has been shewn
equal to the difference ariiing from subtracting tlie logarithm of the divisor u2
from the logarithm of the dividend iz; while the latter is the quotient arising from
dividing the logarithm of ut by the logarithm of u2.