a number between 100 and 1000 is 2 plus a decimal. The logarithm of a number is thus seen to consist, in general, of two parts, an integral part and a decimal part. The integral part is called the characteristic of the logarithm; the decimal part is called the mantissa. The results of our observations may be summarized thus:

Whence we formulate the law: The characteristic of the logarithm of a number is one less than the number of digits in the integral part of the number.

On the other hand, we observe from the table of this article that if a number contain no integral digits, that is, if it be purely decimal, its logarithm is negative. The characteristic in this case can be got by counting the number of zeros before the first significant figure, prefixing the minus sign. It is usual, and better, however, except for special purposes, not to write the characteristic of the logarithm of a decimal number in the form just stated, for reasons which will now be pointed out.

31. The Mantissa. In the common system the mantissa of the logarithm of a number can be made to depend only upon the sequence of digits in the number, and be independent of the position of the decimal point. Let us assume that we know the logarithm of 1.285 to be 0.1089. It follows, multiplying successively by ten, that

which verifies the law we have stated. If, however, we divide 100.1089 successively by 10 we find

This is the true form of the logarithm of a purely decimal number, and for certain purposes this is the form which must be used.*

It is obvious from the preceding discussion that the mantissa corresponding to a given sequence of digits remains the same as long as the sequence contains one or more integral digits, but that as soon as the sequence is a purely decimal number the mantissa changes. To obviate this difficulty and to keep the mantissa the same for a given sequence of digits regardless of the position of the decimal point, we note that the number - 0.8911 may be written, without change of value, in the form 9.1089-10. We have added 10 and subtracted 10, and have therefore left the value unchanged. We may then say

and if we agree to use the latter form† we see that the mantissa of the logarithm of .1285 (that is, 1089) is the same as the mantissa of the logarithm of the sequence 1285 when it contains integral digits. We may now write

log 1.285=0.1089
log 12.85 = 1.1089

log.1285 = 9.1089 - 10
log .01285 = 8.1089

=8.1089-10

and make the statement: In the common system the mantissa of a logarithm is unique for a given sequence of digits. The

* For example, in dividing one logarithm by another.

† This form, 9.1089-10, is perfectly convenient as long as the operations to be performed are addition and subtraction, which are the usual operations in dealing with logarithms.

D

characteristic is one less than the number of integral digits. If a number be purely decimal, count the decimal point and the zeros before the first significant figure. The result subtracted from 10 minus 10 will be the characteristic.

32. Four Computation Theorems. The use of logarithms in computation depends upon the four following theorems:

I. In any system the logarithm of a product is equal to the sum of the logarithms of its factors.

To prove, log mn ••• s = logam + loga n +

+ log, s.

This theorem replaces the operation of multiplication by the simpler operation of addition.

II. In any system the logarithm of a quotient is equal to the logarithm of the dividend minus the logarithm of the divisor.

This theorem replaces the operation of division by the simpler operation of subtraction.

III. In any system the logarithm of a power of a number is equal to the exponent of the power times the logarithm of the number.

This theorem replaces the operation of involution, or successive multiplications, by the simpler operation of a single multiplication.

IV. In any system the logarithm of a root of a number is equal to the quotient of the logarithm of the number by the index of the root.

This theorem replaces the operation of evolution, or extraction of roots, by the simpler operation of division.

Another theorem, important in the theory of logarithms, but of which no application is made in the study of trigonometry is the following:

By means of this theorem the logarithm of a number to any base can be found if the logarithms of numbers to some one base are known. Thus, assuming that logarithms to the base 10 are known,

log. 71.24

1.8527

log 10 71.24 log10 71.24
log10 e log10 2.718 0.4343

4.2659.

As a corollary of the above theorem we have, putting

33. Special Properties of Logarithms. In addition to the preceding theorems we may note the following properties of logarithms:

1. In any system the logarithm of 1 is 0. For, by the definition of zero exponent, ao 1. Therefore, log。 1 = 0. 2. In any system the logarithm of the base is 1.

For a1 = a. Therefore, log, a = 1.