INTRODUCTION, CHAPTER I. OF LOGARITHMS. 1. LOGARITIIMS of numbers are the indices that denote the different powers to which a given number must be raised to produce those numbers. 2. If a be the given number, whose indices and powers are to be considered, then as being put equal to n, a, the given number, or root, is called the base of the system of logarithms, n the number whose logarithm is considered, and Ex, the logarithm of that number. 3. Any number, except 1, may be taken for the base of a system of logarithms. In the system in general use, the base is 10; and this system affords the greatest facilities in calculations, because 10 is the base of the common numeration, both in whole numbers and decimal fractions. 4. Taking a=1=n, we have, Ex=log. n; and putting a+v= m, gives, Ey=log. m. If the equations aʻ=n, and a%=m, be multiplied together, member by member, we have, a? Xay= nxm, or az+y=n Xm. In this expression, x+y is the logarithm of n xm (2); from which we conclude, that the sum of the logarithms of any two numbers, is equal to the logarithm of their product. 5. If the equations a’=n, arm, be divided, member by a" member, ; or a?-=-. In this expression, r-y is the ay n n m n logarithm of " (2); from which we conclude, that, the difference of the logarith.ns of any two numbers, is equal to the logarithm of their quoticni. 6. If in the equation aʻ=n, both members be raised to the mth power, a"i=n". Here, mx is the logarithm of nm; from which it appears, that the logarithm of the power of any number, is equal to the logarithm of that number, multiplied by the index of that power. 7. If the mth root of both members of the equation aʻ=n, 1 m be taken, then, a =n"; but is the logarithm of nm; from which it appears, that the logarithm of the root of any number, is equal to the logarithm of that number divided by the index of the root. 8. It is evident, that the results obtained in the last four articles are equally true, whether the logarithms be positive or negative. These results show, that the addition of logarithms corresponds to the multiplication of their numbers, the subtraction of logurithms, to the division of numbers; their multiplication, to the ruising of powers ; and their division, to the extraction of Tools, 9. Returning to the equation atr=n, in which Ex=log. 1, and applying it to the common system, in which the base is 10, we have, (10)* : (10)*:(10)*: (10) : (10)° :(10)-!:(10)-9 :(10)-3: (10)-4 100101: 1000 : 100 i 10 : 0.1 : 0.01 :0.001 : 0.0001 num. :-3 : -4 log. Unity being the number which divides the whole numbers from the decimal fractions, we shall begin with it, and explain some properties of the logarithms of whole numbers. The logurithu ot' l is (); and this is the case in all systems, for whatever be the base, its 0 power is 1 : but the index of the base is the logarithm of the power; therefore, 0 is the loga . from unity upwards, the logarithms of all numbers, which are greater than 1, and less than 10, are greater than 0, and less than 1 : their values are generally expressed by decimal fractions; thus, the log. 2=0.301030. The logarithms of numbers greater than 10, and less than 100, lie between 1 and 2, and are generally expressed by unity and a decimal fraction: thus, the log. 50=1.698970. The logarithms of numbers greater than 100, but less than 1000, are greater than 2, and less than 3, and are expressed by uniting 2 with a decimal fraction : thus, the log. 126= 2.100371. The whole number on the left of the decimal point is called the characteristic, or index of the logarithm. The number of units which it contains, is always one less than the number of places of figures in the number whose logarithm is taken. Thus, in the first case, for numbers between 1 and 10, there is but one place of figures, and the characteristic is 0. In the second case, for numbers between 10 and 100, there are two places, and the characteristic is 1. In the third case, for numbers between 100 and 1000, there are three places, and the characteristic is 2; and in like manner for any num. ber of places whatsoever. TABLE OF LOGARITHMS, 10. If the logarithms of all the numbers between 1 and any given number, be calculated and arranged in a tabular form, such table is called a table of logarithms. The table annexed shows the logarithms of all numbers between 1 and 10,000. 11. The first column, on the left of each page of the table of logarithms, is the column of numbers, and is designated by the letter N; the logarithms of these numbers are placed directly opposite them, and on the same horizontal line. 12. To find, from the table, the logarithm of any whole number. If the number be less than 100, look on the first page of the table of logarithms, along the column of numbers under N, until the number is found; the number directly opposite it, in the column designated Log., is the logarithm sought. Find, in the column of numbers, the first three figures of the given number. Then, pass across the page, along a horizontal line, into the columns marked 0, 1, 2, 3, 4, &c., until you come to the column which is designated by the fourth figure of the given number: to the four figures so found, two figures taken from the column marked 0, are to be prefixed. If the first four figures found stand opposite to a row of six figures in the column marked 0, the two figures from this column, which are to be prefixed to the four before found, are the first two on the left hand ; but, if the first four figures are opposite a line of only four figures, you are then to ascend the column, till you come to the line of six figures: the two figures at the left hand are to be prefixed, and then the decimal part of the logarithm is obtained; to which prefix the characteristic (9), and you have the logarithm sought. In several of the columns designated 0, 1, 2, 3, 4, 5, &c., small dots are found. In such cases, a cipher must be written for each of those dots; and the two figures, from the first column, which are to be prefixed, are found in the horizontal line directly below. Thus, the log. 2189 is 3.340047, the two dots being changed into two ciphers, and the 34 from the column 0, prefixed. The two figures from the column 0, must also be taken from the line below, if any dots shall have been passed over, in passing along the horizontal line: thus, the logarithm of 3098 is 3.491081, the 49 from the column 0, being taken from the line 310. 14. If the number exceed 10,000, or consist of five or more places of figures, consider all the figures after the fourth from the left hand, as ciphers. Find from the table, the logarithm of this number, which will be the same as the logarithm of the first four places, excepting the characteristic. Take from the last column on the right of the page, marked D, the number on the same horizontal line with the logarithm, and multiply this number by the numbers that have been considered as ciphers : then, cut off from the right hand as many places for decimals as there are figures in the multiplier, and add the product, so obtained, to the first logarithm, for the logarithm sought. log. of 672800 is found, on the 11th page of the table, to be 5.827886, by prefixing the characteristic 5. The number corresponding in the column D is 65, which being multiplied by 87, the figures regarded as ciphers, gives 5655; then, pointing off two places for decimals, the number to be added is 56.55. This number being added to 5.827886, gives 5.827942 for the logarithm of 672887; the decimal part, .55, being omitted. This method of finding the logarithms of numbers from the table, supposes that the logarithms are proportional to their respective numbers, which is not rigorously true. In the example, the logarithm of 672800 is 5.827886; of 672900, a number greater by 100, 5.827951: the difference of the logarithms is 65. Now, as 100, the difference of the numbers, is to 65, the difference of their logarithms, so is 87, the difference between the given number and the least of the numbers used, to the difference of their logarithms, which is 56.55: this difference being added to 5.827886, the logarithm of the less number, gives 5.827942 for the logarithm of 672887. The use of the column of differences is therefore manifest. 15. The logarithm of a fractional number is easily found, from what has already been said. If the fractional number exceed unity, as , its logarithm is equal to the log. 136 - log. 25 (5). If it be less than unity, as is, its logarithm may be written under two different forms. First, the log. Mas =log. 15 -- log. 125=--(log. 125 --- Jog. 15)=-(2.096910 -1.176091)=-0.920819; the number 0.920819 being en = tirely negative. In the equation log. 15—log. 125=-0.920819, if the log. 125 be transposed to the second member, the log. 15=log. 125 -0.920819. Let N' be the number whose logarithm is -0.920819, and N the number whose logarithm is +0.920819; then, the log. 15 - log. 125=log. N. Since the difference of logarithms of the two numbers is equal 125 to the logarithm of their quotient (5), the log. 15=log. N: 136 1 |