EXPLANATION OF THE TABLES.

1. Definitions and Rules. If three numbers n, a, x have such values that the equation

is true, then x is called the logarithm of n to the base a. If, without changing a, we give to n and x all possible values, consistent with this equation, the values of x thus obtained form a system of logarithms to the base a.

Hence:The logarithm of a number to a given base is the exponent of the power to which the base must be raised to produce the number.

Suppose 9 is taken for the base, then

log 81 2, because 92

THITH

1

81 729

*

3

9

90 = 1

In every system the logarithm of the base is 1, and the logarithm of 1 is 0. This follows directly from the definition, or from (1); for if na, x must be 1; and if n= 1, x must be 0, without respect to the value of a.

It is plain, since any number will serve as the base of a system of logarithms, that the number of such systems is indefinite.

vii

The systems of logarithms commonly used are:

(1.) The common or Briggian* system, with the base 10. (2.) The natural or Napieriant system with the base e = 2.7182818285....

defined by the convergent infinite series

Of these two systems, the first is used for all purposes of numerical computation, and the second for purely analytical purposes.

The logarithms of these tables (except in Table VII.) are common or Briggian logarithms.

The corresponding logarithms of any two systems are in a constant ratio to each other. Thus the relation between common and Napierian logarithms is

(This equation is read: "Logarithm of n to the base 10 equals the reciprocal of the logarithm of 10 to the base e, multiplied

by the logarithm of n to the base e.") The factor

1

log, 10

is

called the modulus of the common system. It is represented by M, and its value to ten places is 0.4342944819.

The rules governing the use of logarithms in computation are the following:

I. To multiply numbers, find the logarithm of each factor, and add them; the sum is the logarithm of the product.

II. To divide one number by another, subtract the logarithm of the divisor from the logarithm of the dividend; the difference is the logarithm of the quotient.

III. To raise a number to any power multiply the logarithm of the number by the exponent of the power; the product is the logarithm of the required power of the number.

* Named for Henry Briggs (1556–1631), who first suggested the use of the base 10.

+ Named for John Napier, Baron of Merchiston, in Scotland (15501617), the inventor of logarithms.

IV. To extract any root of a number, divide the logarithm of the number by the index of the root; the quotient is the logarithm of the required root of the number.

These statements and rules are given without proof, as the purpose here is simply to familiarize the student with the mechanism and use of the tables. The theory of logarithms is set forth in text-books on algebra, to which the student is referred. In the same place will be found an explanation of how logarithms are computed.

TABLE I. Common Logarithms of Numbers. (Pages 1–19.)

2. Characteristic and Mantissa. A logarithm consists, usually, of two parts: a whole number, called the characteristic, and an incommensurable decimal fraction, called the mantissa. The table gives only the mantissa; the characteristic, which may be positive, negative, or zero, must be supplied in every case by the computer. The mantissa is always positive, except in the logarithms of exact powers of 10, when it is zero. Since 10 is the base we have:

This series of equations can be extended indefinitely in both directions.

=

Let us now consider two numbers which contain the same sequence of figures, with different positions of the decimal point, say 72.936 and .72936. Now 72.936 100 × .72936. Hence, by Rule I, § 1 log 72.936 = log 100+ log .72936, or, by (a) =2log .72936. Hence, since any change in the position of the decimal

point in a number is equivalent to multiplication or division by a power of 10, the effect produced upon the logarithm of the number by a change of this kind is to increase it or diminish it by a whole number; that is, the characteristic is affected by such a change, but not the mantissa. We have, therefore, the following important fact:

I. The mantissa of the logarithm of a number depends only upon the sequence of figures in the number.

Referring again to (a), we note that for all numbers greater than 1 and less than 10 (all numbers with one significant figure before the decimal point) the logarithm is greater than O and less than 1, that is, its characteristic is 0; for all numbers greater than 10 and less than 100 (all numbers with two significant figures before the decimal point) the logarithm is greater than 1 and less than 2, that is, its characteristic is 1; for all numbers greater than 100 and less than 1000 (all numbers with three significant figures before the decimal point) the logarithm is greater than 2 and less than 3, that is, its characteristic is 2; and so on. Hence, we have the following rule:

II. The characteristic of the logarithm of a number greater than unity is one less than the number of significant figures preceding the decimal point.

Again, from (a) it will be seen that if a number is greater than .1 and less than 1, its logarithm is between 0 and -1; that is, using a positive mantissa, which we always do, it is -1 the mantissa, hence the characteristic is -1; if the number is greater than .01 and less than .1, the logarithm is between 1 and 2, which is written —2 + the mantissa, that is, the characteristic is -2; if the number is greater than .001 and less than .01, the logarithm is between 2 and -3, which is written -3+ the mantissa, that is, the characteristic is —3, and so on. Hence, we have the following rule:

III. The characteristic of the logarithm of a number less than unity is negative, and is numerically one greater than the number of ciphers between the decimal point and the first significant figure.