OF LOGARITHMS. 1. Logarithms are a kind of artificial numbers, invented by Lord Napier, of Merchiston Castle, near Edinburgh, to facilitate certain calculations, such that while the natural numbers increase in geometrical progression, their logarithms increase in arithmetical progression. By this artifice the operation of multiplying numbers is reduced to that of adding their logarithms; that of dividing numbers to that of subtracting their logarithms; the raising of powers to that of multiplying the logarithms; and the extracting of roots to that of dividing the logarithms of the numbers. 2. If n=a*, x is called the logarithm of the number n to the base a; or the logarithm of a number to a given base is that power of the base which is equal to the given number. 3. The base may be any number whatever except 1, but the most convenient base is 10, which is the base of the common logarithms given in this work; but a simple illustration of the nature of logarithms may be given, by taking the base 2, or 3, or 10, and only using those numbers of the natural series whose logarithms are whole numbers. 4. 5. 6. 7. 0. 1. 2. 3. 8. log. 1. 10. 100. 1000. 10,000. 100,000. 1,000,000. 10,000,000. 100,000,000 nat. num. In either of the above series it will be found, that if two logarithms be added together, such as 3 and 5, their sum 8 points out the number which is the product of the numbers of which 3 and 5 are the logarithms; or if the difference of two logarithms, as of 7 and 5, be taken, the remainder, 2, will point out the quotient arising from dividing the number of which 7 is the logarithm, by that of which 5 is the logarithm; hence, B 4. The sum of the logarithms of two numbers is the logarithm of their product, and the difference of the logarithms of two numbers is the logarithm of their quotient; which may be demonstrated generally as follows, by the notation in (Art. 2); for let na, and n'a, then their product gives nn'=a*a*=a(x+), where x+x', or the sum of the logarithms of the numbers, is evidently the logarithm of their product: also their quotient gives n ax =a(-s) where x-x', or the difference of the logarithms of the numbers, is the logarithm of their quotient. 5. The logarithm of a power or root of any number is the logarithm of the number multiplied by the exponent which indicates that power or root; for let na represent any number having its logarithm x to the base a, and let m represent the exponent of any power, (m= an integer), or root, (m a fraction); then we have n"=(a*) "=aTM, therefore if x be the logarithm of n, ma is the logarithm of its mth power or root; hence by giving proper values to m, we have the following rules for raising numbers to any power, or extracting any root. THI 6. To square any number by logarithms, multiply the logarithm of the number by two, and the product will be the logarithm of its square. 7. To find the cube of any number by logarithms, multiply the logarithm of the number by three, and the product will be the logarithm of its cube. 8. To find the square root of any number by logarithms, divide the logarithm of the number by two, and the quotient will be the logarithm of its square root. 9. To extract the cube root of a number by logarithms, divide the logarithm of the number by three, and the quotient will be the logarithm of its cube root. NOTE. The method of calculating logarithms is fully explained and illustrated by examples, in the Supplement to the Key to this work. DESCRIPTION OF THE TABLES OF LOGARITHMS. 10. Since the base of the logarithms here used is 10, any number will be equal to 10, where x is the logarithm of the number; if x=0, then 10o-1, therefore the logarithm of 1 is 0; if x=1, then 10'-10, the logarithm of 10 is 1, hence the logarithms of all numbers between 1 and 10 are greater than 0 and less than 1, that is, they are decimal fractions. In the same manner, by making x=2, it is evident that the logarithm of 100 is 2, and therefore that all numbers between 10 and 100 will have their logarithms 1, with a decimal fraction annexed; but all numbers between 10 and 100 are written with two figures, hence the integral part of the logarithm is one less than the integral places in the number, and this rule may in the same manner be proved universal. For this reason the decimal part (or mantissa) only of the logarithms is inserted in the tables, and the integral part (or index) is prefixed by the following rule: 11. Find how many places from the unit's figure the first significant figure of the number is, that prefixed to the decimal part found from the tables will be the true logarithm. If the first significant figure be to the left of the unit's place, the index is plus, but if to the right, the index is minus, and the sign (-) must be placed over it; thus the logarithm of 150 is 2·176091, while that of 0·015 is 2.176091, where the (—) applies only to the index, not to the mantissa, it being always (+). 12. To find the logarithm of any whole number under 100. Look for the number under N, in the first page of the logarithms, then immediately on the right of it is the logarithm sought, with its proper index. Thus the log. of 56 is 1-748188, and the log. of 91 is 1.959041. 13. To find the logarithm of any number between 100 and 1000. Find the given number in some of the following pages of the table, in the first column, marked N, and immediately on the right of the number stands the decimal part of the logarithm, in the column marked 0, at the top and bottom, to which decimal prefix the proper index, (Art. 11). Thus the log. of 457 is 2-659916, and the log. of 814 is 2.910624. 14. To find the logarithm of any number consisting of four places. Find the first three figures in some of the left-hand columns of the pages, as in the last case, and the fourth figure at the top or bottom of the table; then directly under the fourth figure, and in the straight line across the page, from the first three figures of the number will be found the decimal part of the logarithm sought, and the index must be supplied by (Art. 11). Thus the log, of 7.384 is 0.868292, the log. of 793-5 is 2-899547, and that of ⚫04638 is 2.666331. 15. To find the logarithm of any number consisting of five or six places. Find the logarithm of the first four left-hand figures, as in the last article, to which prefix the proper index by (Art. 11); then from the right-hand column, marked D, (meaning tabular difference), take the number opposite to that logarithm, and multiply it by the remaining figures of the natural number; point off from the right-hand side of the product as many figures as there are in the multiplier; then add the rest of the product to the logarithm before found, and the sum is the logarithm required. Ex. 1. Required the logarithm of 34863. Diff. 125 ×3=37·5, therefore add Gives log. of 34863 4.542327 37 =4.542364. NOTE. It may be remarked here, that when the first figure pointed off in the product of the difference is 5, we may either take the remaining figures, or the remaining figures increased by one, as in either case the error is, in the last figure of the logarithm; but if the figure or figures pointed off exceed 5, the remaining figures must always be increased by unity, as the error by this means will be less than, in the last figure of the logarithm, whereas by omitting them, it would be greater than a half. Ex. 2. Required the logarithm of 46.8375. Diff. 93 × 75=69.75, .. add Hence log. of 46-8375 1.670524 =1.670594. Ex. 3. Required the logarithm of '00764525. Log. of 007645 by the table (Art. 11) is 3-883377 14 16. To find the logarithm of a vulgar fraction, or of a mixed number. Reduce the vulgar fraction to a decimal; then find the decimal part of its logarithm by the preceding rules, and prefix the proper index, as found by (Art. 11.) Or, from the logarithm of the numerator subtract the logarithm of the denominator, and the remainder will be the logarithm of the fraction sought, (Art. 4.) A mixed number may be reduced to an improper fraction, and its logarithm found in the same manner. Ex. 1. Required the logarithm of Take log. of 24 Hence log. of, or ·2916, is =2916. =1.380211 =1.464887. NOTE. If the logarithm of this example be taken out from the decimal fraction, since it repeats 6, which is the decimal of 3, we must add of the tabular difference to the logarithm of the first four figures, in order to obtain the true logarithm; thus log. of *2916=1·464788, and the tabular difference is 149, two-thirds of which is 99, which being added to the log. formerly found, gives 1-464887, the same as before. Ex. 2. Required the logarithm of 231-23-25. Take log. of 4 Hence log. of 23, or 23.25 =1.968483 =0.602060 =1.366423. 17. To find the natural number answering to any given logarithm. Look for the decimal part of the given logarithm in the different columns, until you find either it exactly, or the next less. Then in a line with the logarithm found, in the left-hand column marked N, you will have three figures of the number sought, and on the top of the column in which the logarithm found stands, you have one figure more, which annex to the other three; place the decimal point so that the number of integral figures may be one more than the units in the index of the logarithm, (Art. 11), and if the logarithm was found exactly, you have the number required. If the logarithm be not found exactly in the table, subtract the logarithm next less than it found in the table from the given one, and divide the remainder, with two ciphers annexed, by the tabular difference, which will give other two figures, (if the quotient give only one figure, the first is a cipher), which annex to the four figures found by inspection, and it will be the number sought, except the index show that the number consists of more than six figures, |