LOGARITHMS.

Logarithms are a set of numbers, so contrived that the products in multiplication, and the quotients in division, are obtained by means of addition and subtraction only.

Or, Logarithms are a series of numbers in arithmetical progression, corresponding to another series in geometrical progression, the arithmetical series being the indices or powers of a given quantity, as a base. *

As 10 is the base of our present system of arithmetic, so is it also employed as the base of the logarithms generally used. On this scale all the common tables of logs. are constructed. If we assume a series of numbers in geometrical progression, proceeding from 1, the ratio being 10, and set over them a series of numbers in arithmetical progression, beginning with 0, the common difference being 1, the numbers in the arithmetical series will be the logs. of the corresponding numbers in the geometrical series. Thus―

0, 1, 2, 3, 4, 5, 6,

7,

8.

1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000.

From this it appears that the numbers in the arith

* The invention of Logarithms is due to Lord Napier, Baron of Merchiston, in Scotland. In 1614, the inventor published the first tables of these numbers. His Logarithms are of that form which we call Hyperbolic Logarithms. In this system 1 is the logarithm of 2.718282. We are indebted for the modern table to Brigg, in whose system the logarithm of 10 is 1. The modern tables most in use are those of Tayler's and Doctor Hutlon's.

Various methods have been proposed to construct logarithms, but the most simple and expeditious is that which may be seen in Gregory's Philosophy of Arithmetic, which see.

metical series, which are the logs. of the corresponding numbers beneath them, are the exponents of the different powers of 10. But the sum of two exponents is the exponent of the product arising from the multiplication of their corresponding natural numbers, (see Philosophy of Arithmetic.) Hence it appears, that if the logs. of any two numbers be added together, the sum will be the log. of the product of these numbers.

In the foregoing series, if we add 2 and 4, the sum 6 will be the log. of 100× 10000, (=1000000.) Now as the addition of any two logs. answers to the multiplication of their corresponding natural numbers, it is evident that the converse will hold true; that is, that the subtraction of any two logs. will answer to the division of their corresponding natural numbers. Resuming the last example, if we take 4 from 6, the remainder, 2, will be the log. of 1000000÷10000 (=100.)

Again, as the sum of any two logs. is the log of the product of their corresponding natural numbers, if we suppose the two numbers equal, then double the log. of one of them must be the log. of their product, which is the second power of one of them; thus 1+1 (=2), is the log. of 10×10 (=100); and for the same reason 1+1+1 (=3), is the log. of 10x 10x 10 (=1000.) Hence it is evident, that twice the log. of any number is the log. of the second power of that number; three times the log. of any number, will be the log. of the third power of that number, &c. Therefore the log. of any power of a number, is equal to as many times the log. of the number as is denoted by that power. Thus, the log. of 104 is 1x 4 (=4); the log. of 1002 is 2x 2(4), &c. Then, to find the second, third, or

fourth power of any number, multiply its log. by 2, 3, or 4, according as the case may require; thus, to find the third power of 100, or (100)3, we multiply the log. of 100, which is 2, by 3, and the product 6, is the log. of (100)3, and the corresponding number to the log. 6, which is 1000000, is the third power of 100, or (100)3.

The converse of this is true; that is, that to extract the square root of any number, we divide the log. of that number by 2, and the number answering to the quotient is the root required. To extract the cube root of any number, we divide the log. of the number by 3, and the number answering to the quotient is the root required. Thus, to extract the square root of 10000, or (10000), the log. of 10000 is 4, which being divided by 2, gives 2, which is the log. of 100, the root required. In like manner, if we require to extract the cube root of 1000000, or (1000000), the log. of 1000000 is 6, which being divided by 3, gives 2, which is the log. of 100, the root required.

As the logs. of 1, 10, 100, 1000, &c. are 0, 1, 2, 3, &c. respectively, it is evident that the log. of any number falling between 1 and 10 will be 0 and some decimal parts; that of any number between 10 and 100, 1 and some decimal parts; of any number between 100 and 1000, 2 and some decimal parts; and so on for higher numbers. Hence, the index or characteristic of any log. is always 1 less than the number of figures in the integral part of the natural number.

Again, as a proper fraction is an expression arising from the division of the numerator by the denominator, and as this division is equal to the subtraction of their corresponding logs., it is obvious, that the log. of

a proper fraction will have a negative index; thus the log. of 10, or .1, is found by subtracting the log of 100, which is 2, from the log. of 10, which is 1, the remainder is -1; in like manner, the log. of .01 is -2; that of .001, -3; that of .0001, -4; and so on for other decimals.

As the index or characteristic may be easily known by the computer, it is usually omitted in tables of logs. Most tables contain the logs. of the natural numbers from 1 to 10000, generally to six places of figures.

In some tables, the logs. are continued to more decimal places.

To find the log. of any number, consisting of three places of figures; suppose 123.

Look in the left hand column for the number; then .08990 in the second column is the decimal part of its log., and the index is 2, as 123 consists of three integers; hence the log. of 123 is 2.08990.

To find the log. of any number consisting of four places of figures; suppose of 2157.

Look in the left hand column for the three first figures as before, that is 215; then under 7, at the top of the table, and in a horizontal line with 215, will be found 33385; and the index is 3, the integer consisting of four figures therefore the log. of 2157 is 3.33385.

To find the log. of any number consisting of five figures; suppose 24676.

Find the log. of the four left hand figures, as in the last case, which deduct from the next log. greater;

then say, as 10 is to the difference, so is the fifth figure of the given number, to a fourth, which added to the less of the two logs., will give the log. sought.

As 10 17 6: 10.2, the 4th number.

The difference of the logs. is found in the right hand column of some tables, which saves the trouble of subtraction.

The above stating is founded on the supposition, that the differences of logs. are as the differences of the corresponding numbers.

Thus,

4.39217

4.39234

10 diff. of numbers.

17 diff. of logs.

Then 10: 17 :: 6 : 10.2

To find the log. of any number consisting of six places of figures.

Find the log. of the first four figures, as before, which deduct from the next log. greater; then say, as 100 is to the difference, so are the two remaining figures to a fourth number, which added to the less of the two logs., will give the log. sought.