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

5

and therefore the logarithm of u lies between n-1 and n, and consequently consists of n-1 units increased by some decimal: that is,
the characteristic of log,0 u is n-1.
Next, let uf be a decimal having n-1 ciphers between the decimal
point and the first significant figure.
'This decimal uf lies between - and 0, or 10-(-) and 10-",
104-1 j',.*. the logarithm of u' lies between -(n-1) and -n, and consequently consists of - n, increased by some positive decimal part: that
is, the characteristic of log,,o' is - n.
Hence the general rule. For numbers wholly or partly integral, the
characteristic is always less by unity than the number of integral
places of which the number consists: and for decimals, the characteristic is the number (taken negative) which expresses the distance of
the first significant figure of the decimal from the place of units.
8. PROP. Thle logarithmn of a numnber less than 1, being negative, can
Oalways be expressed so that its mzantissa shall be positive, and only its characteristic negative.1
Let u be a number less than 1,
n the characteristic, m the mantissa of its logarithm.
Then log,0u= - (n + n) = -n - m + 1 - 1
- (n + 1) + (1 - 9m) of which 1 - m is positive.
Hence a logarithm wholly negative may be transformed into one
whose characteristic only is negative, by increasing the negative
characteristic by 1 and replacing the mantissa by its arithmetic complement, or its defect from 1.
And conversely. A logarithm whose characteristic only is negative,
may be transformed into a logarithm wholly negative by diminishing
the characteristic by 1, and replacing the mantissa or decimal part by
its arithmetical complement.
partly integral and partly decimal, as will be seen in the logarithms of the numbers
composed of the significant digits 6375.
Numbers. Logarithms. Numbers. Logarithms.
6375 3.8044802 637'5......... 2-8044802
63750...... 48044802 63-75......... 1-8044802
637500...... 5-8044802 6-375......... 0'8044832
6375000...... 6-8044802 -6375......... 1 8044802
63750000...... 7 -8044802 06375......... 2-8044802
637500000...... 8808044802 006375......... 3-8044802
1 Ex. 1. Log, j 1 } = lgloo 1 - loo 2 = - loglo 2 = -3010300
= -1 + (1 —3010300)
= 1'6989700, a logarithm with its mantissa positive and its
characteristic negative.
Ex. 2. Loglo i, =- logo 5 —logo 9, which is wholly negative.
= (1 + logio 5) -l -logo 9 = (logio 10 + logio 5) 1 -logo 9
= + (log10 50 —olog 9).