356 ALGEBRA.
(2) The mantissæ are the same for the logarithms of all numbers which have the same significant digits; so that it is sufficient to tabulate the mantissæ of the logarithms of integers.
This proposition we proceed to prove.
439. Let N be any number, then since multiplying or dividing by a power of 10 merely alters the position of the decimal point without changing the sequence of figures, it follows that N x 10^p, and N 10^p, where p and q are any integers, are numbers whose significant digits are the same as those of N.
Now log (N x 10^p)=log N+ p log10 =log N+ p (1).
Again, log (N 10^q)= log N- qlog10 =log N-q (2).
In (1) an integer is added to log N, and in (2) an integer is subtracted from log N; that is, the mantissa or decimal portion of the logarithm remains unaltered.
In this and the three preceding articles the mantissæ have been supposed positive. In order to secure the advantages of Briggs’ system, we arrange our work so as always to keep the mantissa positive, so that when the mantissa of any logarithm has been taken from the Tables the characteristic is prefixed with its appropriate sign, according to the rules already given.
440. In the case of a negative logarithm the minus sign is written over the characteristic, and not before it, to indicate that the characteristic alone is negative, and not the whole expression. Thus 4.30103, the logarithm of .0002, is equivalent to -4 + .30108, and must be distinguished from -4.30103, an expression in which both the integer and the decimal are negative. In working with negative logarithms an arithmetical artifice will sometimes be necessary in order to make the mantissa positive. For instance, a result such as -3.69897, in which the whole expression is negative, may be transformed by subtracting 1 from the characteristic and adding 1 to the mantissa. Thus,
-3.69897 = -4 +(1 - .69897) = 4.30103.
