Logarithms may be defined as a series of numbers in _arithmetical_
progression, as 0, 1, 2, 3, 4, etc., which bear a definite relationship
to another series of numbers in _geometrical_ progression, as 1, 2, 4,
8, 16, etc. A more precise definition is:—The logarithm of a number to
any base, is the _index of the power_ to which the base must be raised
to equal the given number. In the logarithms in general use, known as
_common logarithms_, and with which we are alone concerned, 10 is the
base selected. The general definition may therefore be stated in the
following modified form:—_The common logarithm of a number is the index
of the power to which 10 must be raised to equal the given number._
Applying this rule to a simple case, as 100 = 10^2, we see that the base
10 must be squared (_i.e._, raised to the 2nd power) in order to equal
100, the number selected. Therefore, as 2 is the index of the power to
which 10 must be raised to equal 100, it follows from our definition
that 2 is the common logarithm of 100. Similarly the common logarithm of
1000 will be 3, while proceeding in the opposite direction the common
log. of 10 must equal 1. Tabulating these results and extending, we
have:—
Numbers 10,000 1000 100 10 1
Logarithms 4 3 2 1 0