Logarithms are series of artificial numbers, so arranged with reference to a set of natural numbers that the addition of the logarithms shall correspond with the multiplication of the natural numbers belonging to them; and subtraction of logarithms answers for division; while involution, or the raising of powers, is performed by the multiplication of logarithms; and evolution, or the extraction of roots, by the division of logarithms.

To illustrate this, let us take -

 For Natural Numbers the Geometrical Series . . 1 10 100 1000 10000 100000 1000000 And for their Logarithms the Arithmetical Series 0 1 2 3 4 5 6

From this it appears that the log. of 1 is 0, that of 10 is 1, of 100 is 2, etc.; that the log. of any number below 10 is a fraction, above 10 and under 100 is 1; with a fraction, and between 1000 and 100, is 2 with a fraction, and so on. Hence it is evident that the portion of a log. which constitutes the whole number, and is denominated the Index, is always one less than the numbers of figures for which it is the log. This general rule is so easy of application, that the Indexes of Logarithms are never printed in the tables, but left to be supplied by the operator.

The rule for determining the Index descends as well as ascends, and applies with equal facility to numbers below and above unity; but when applied to numbers below unity, it must be distinguished by a negative sign thus -

 NATURAL NUMBER. LOGARITHM. .000001 ........................ 6.0000000 .00001 ........................ 5.0000000 .0001 ........................ 4.0000000 .001 ........................ 3.0000000 .01 ........................ 2.0000000 .1 ........................ 1.0000000 1. ........................ 0.0000000 10. ........................ 1.0000000 100 ........................ 2.0000000 1000 ........................ 3.0000000 etc. ........................ etc.

To furnish the means of illustrating this important subject by a few examples, and to give the reader an opportunity of working cases by logarithms when the numbers to be operated upon are not very large, we subjoin