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 = 10², 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:—

It will now be evident that for numbers

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:—

Combining the two tables, we can complete the logarithms. Thus for 3 multiplied successively by 10, we have:—

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:—

so that for 3 divided successively by 10, we have:—

Here again we see that with the same significant figures in the numbers, the mantissa of the logarithm has always the same (positive) value, but the characteristic is one more than the number of 0’s immediately following the decimal point, and is negative, as indicated by the minus sign written over it. Only the decimal parts of the logarithms of numbers between 1 and 10 are given in the usual tables, for, as shown above, the logarithms of all tenfold multiples or submultiples of a number can be obtained at once by modifying the characteristic in accordance with the rules given.

An examination of the two rows of figures giving the logarithms of numbers from 1 to 10 will reveal some striking peculiarities, and at the same time serve to illustrate the principle of logarithmic calculation. First, it will be noticed that the addition of any two of the logarithms gives the logarithm of the product of these two numbers. Thus, the addition of log. 2 and log. 4 = 0·301 + 0·602 = 0·903, and this is seen to be the logarithm of 8, that is, of 2 × 4. Conversely, the difference of the logarithms of two numbers gives the logarithm of the quotient resulting from the division of these two numbers. Thus, log. 8 − log. 2 = 0·903 − 0·301 = 0·602, which is the log. of 4, or of 8 ÷ 2.

One other important point is to be noted. If the logarithm of any number is multiplied by 2, 3, or any other quantity, whole or fractional, the result is the logarithm of the original number, raised to the 2nd, 3rd, or other power respectively. Thus, multiplying the log. of 3 by 2, we obtain 0·477 × 2 = 0·954, and this is seen to be the log. of 9, that is, of 3 raised to the 2nd power, or 3 squared. Again, log. 2 multiplied by 3 = 0·903—that is, the log. of 8, or of 2 raised to the 3rd power, or 2 cubed. Conversely, dividing the logarithm of any original number by any number n, we obtain the logarithm of the nth root of the original number. Thus, log. 8 ÷ 3 = 0·903 ÷ 3 = 0·301, and is therefore equal to log. 2 or to the log. of the cube root of 8.

Only simple logs. have been taken in these examples, but the student will understand that the same reasoning applies, whatever the number. Thus for 20³ we prefix the characteristic (1 in this case) to log. 2, giving 1·301. Multiplying by 3, we have 3·903 as the resulting logarithm, and as its characteristic is 3, we know that it corresponds to the number 8000. Hence 20³ = 8000.

In this brief explanation is included all that need now be said with regard to the properties of logarithms. The main facts to be borne clearly in mind are:—(1.) That to find the product of two numbers, the logarithms of the numbers are to be added together, the result being the logarithm of the product required, the value of which can then be determined. (2.) That in finding the quotient resulting from the division of one number by another, the difference of the logarithms of the numbers gives the logarithm of the quotient, from which the value of the latter can be ascertained. (3.) That to find the result of raising a number to the nth power, we multiply the logarithm of the number by n, thus obtaining the logarithm, and hence the value, of the desired result. And (4.) That to find the nth root of a number, we divide the logarithm of the number by n, this giving the logarithm of the result, from which its value may be determined.