Logarithms may be defined as a series of numbers in _arithmetical_ progression, as 0, 1, 2, 3, 4, etc., which bear a definite relationship to another series of numbers in _geometrical_ progression, as 1, 2, 4, 8, 16, etc. A more precise definition is:—The logarithm of a number to any base, is the _index of the power_ to which the base must be raised to equal the given number. In the logarithms in general use, known as _common logarithms_, and with which we are alone concerned, 10 is the base selected. The general definition may therefore be stated in the following modified form:—_The common logarithm of a number is the index of the power to which 10 must be raised to equal the given number._ Applying this rule to a simple case, as 100 = 10^2, we see that the base 10 must be squared (_i.e._, raised to the 2nd power) in order to equal 100, the number selected. Therefore, as 2 is the index of the power to which 10 must be raised to equal 100, it follows from our definition that 2 is the common logarithm of 100. Similarly the common logarithm of 1000 will be 3, while proceeding in the opposite direction the common log. of 10 must equal 1. Tabulating these results and extending, we have:— Numbers 10,000 1000 100 10 1 Logarithms 4 3 2 1 0 It will now be evident that for numbers between 1 and 10 the logs. will be between 0 and 1 „ 10 „ 100 „ „ 1 „ 2 „ 100 „ 1000 „ „ 2 „ 3 „ 1000 „ 10,000 „ „ 3 „ 4 In other words, the logarithms of numbers between 1 and 10 will be wholly fractional (_i.e._, decimal); the logs. of numbers between 10 and 100 will be 1 _followed by a decimal quantity_; the logs. of numbers between 100 and 1000 will be 2 followed by a decimal quantity, and so on. These decimal quantities for numbers from 1 to 10 (which are the logarithms of this particular series) are as follows:— Numbers 1 2 3 4 5 6 7 8 9 10 Logarithms 0 0·301 0·477 0·602 0·699 0·778 0·845 0·903 0·954 1·000 Combining the two tables, we can complete the logarithms. Thus for 3 multiplied successively by 10, we have:— Numbers 3 30 300 3000 30,000 etc. Logarithms 0·477 1·477 2·477 3·477 4·477 „ We see from this that for numbers having the _same significant figure_ (or figures), 3 in this case, the decimal part or _mantissa_ of the logarithm is the same, but that the integral part or _characteristic_ is always _one less than the number of figures before the decimal point_. For numbers less than 1 the same plan is followed. Thus extending our first table downwards, we have:— Numbers 1 0·1 0·01 0·001 0·0001 etc. Logarithms 0 −1 −2 −3 −4 „ so that for 3 divided successively by 10, we have:—