1 = 10^0, 2 = 10^(0.3010300), 3 = 10^(0.4771213), ...
10 = 10^1, 11 = 10^(1.0413927), 12 = 10^(1.0791812).
The integral part of a logarithm is called the index or characteristic,
and the fractional part the mantissa. When the base is 10, the
logarithms of all numbers in which the digits are the same, no matter
where the decimal point may be, have the same mantissa; thus, for
example,
log 2.5613 = 0.4084604, log 25.613 = 1.4084604, log 2561300 =
6.4084604, &c.
In the case of fractional numbers (i.e. numbers in which the integral
part is 0) the mantissa is still kept positive, so that, for example,
_ _
log .25613 = 1.4084604, log .0025613 = 3.4084604, &c.
the minus sign being usually written over the characteristic, and not
before it, to indicate that the characteristic only, and not the whole
expression, is negative; thus
_
1.4084604 stands for -1 + .4084604.
The fact that when the base is 10 the mantissa of the logarithm is
independent of the position of the decimal point in the number affords
the chief reason for the choice of 10 as base. The explanation of this
property of the base 10 is evident, for a change in the position of the
decimal points amounts to multiplication or division by some power of
10, and this corresponds to the addition or subtraction of some integer
in the case of the logarithm, the mantissa therefore remaining intact.
It should be mentioned that in most tables of trigonometrical
functions, the number 10 is added to all the logarithms in the table in
order to avoid the use of negative characteristics, so that the
characteristic 9 denotes in reality ~1, 8 denotes ~2, 10 denotes 0, &c.
Logarithms thus increased are frequently referred to for the sake of
distinction as _tabular logarithms_, so that the tabular logarithm = the
true logarithm + 10.
In tables of logarithms of numbers to base 10 the mantissa only is in
general tabulated, as the characteristic of the logarithm of a number
can always be written down at sight, the rule being that, if the number
is greater than unity, the characteristic is less by unity than the
number of digits in the integral portion of it, and that if the number
is less than unity the characteristic is negative, and is greater by
unity than the number of ciphers between the decimal point and the first
significant figure.