terms of the geometrical progression are obtained by division, and those of the arithmetical progression by subtraction. The same observations apply to logarithms when they are 1 fractions, thus if r" denote any number, then will 1.. 2 3 pn. rn rn rn rn, &c. constitute an increasing series of numbers in geometrical progression, of which the indices (B) Considering logarithms as indices to a series of numbers in geometrical progression, where r and n may represent any numbers whatever, it follows that there may be as many different kinds of logarithms as there can be taken different sorts of geometrical series: numbers in very different progressions may likewise have the same logarithms, and on the contrary, the same geometrical series may have different series of logarithms corresponding to them, but in every case the logarithm of 1 is 0. (C) The tables of logarithms in common use are constructed upon a supposition that r= 10; hence it appears that the logarithm of any number whatever is the index of some power of 10. Thus, the logarithm of 10 is 1, being the index of 101; the logarithm of 100 is 2, being the index of 102; the logarithm of 1000 is 3, being the index of 103; the logarithm of 10000 is 4, being the index of 104, &c. Hence the logarithms of all numbers between 1 and 10 will be greater than 0 and less than 1, that is, they will be decimals; between 10 and 100 they will be greater than 1 and less than 2, that is, they will be expressed by 1 with decimals annexed; between 100 and 1000 the logarithms are expressed by 2 with decimals annexed; between 1000 and 10000 they are expressed by 3 with decimals annexed. of 10 -3 و 1 1000 &c. Hence the logarithms of all numbers decreasing from 1 to 1 will be expressed by-1, with decimals annexed; decreasing from 1 to 01, they will be expressed by -2, with decimals annexed; decreasing from 01 to 001, by -3, with decimals annexed, &c. - And, universally, if 10a, then x is the logarithm of a; where a may be any number from 1 to 101,000, the extent of our best modern tables. 10 If 10x*=2, then x=0·30103 the log. of 2. If 10=50, then x=1.69897 the log. of 50. If 10 94261, then x=4·97433 the log. of 94261. The logarithm of 9'4261 97433 = x, viz. 10x = multipying by 10, we divide by 10 in the equation *The design of this chapter is to show the nature and properties of logarithms, and not the different methods of constructing them. Then 10 =10 as far as you please. (D) Hence it appears that all numbers whatever, whether they be whole numbers, mixed numbers, or decimals, have the decimal part of their logarithms the same, the only difference being in the index or whole number; and that the index to the logarithm of a whole number is affirmative, and the index to the logarithm of a pure decimal is negative; but, in every case, the decimal part of the logarithm is affirmative. CHAP. II. THE USE OF THE TABLE OF LOGARITHMS. PROPOSITION I. (E) To find the logarithm of any whole number, or mixed decimal, consisting of one, two, three, or four figures. This proposition will appear plain from the following examples, observing that the decimal parts only of the logarithms are put down in the tables; because the index, or whole number, of the logarithm is always an unit less than the number of figures contained in the natural whole number to which it belongs. (C. 2.) Required the logarithm of 5. Look in the table in the column marked No. for 5, and against it in the column log. stands 69897. The natural number containing but one figure, the index is therefore 0. Required the logarithm of 74. Look for 74 in the column marked No. and against it stands ⚫86923. The natural number containing two figures, the index is 1, therefore the logarithm of 74 is 1.86923. Required the logarithm of 743. Look for 743 in the column marked No. and against it stands 87099. The natural number containing three figures, the index is 2, therefore the logarithm of 743 is 2.87099. Required the logarithm of 7438. Look for 743 in the column marked No. as before, and directing your eye to the numbers in the top column of the table, find the remaining figure 8, guide your hand down that column till you come in the same line with 743, against which you will find 87146. The natural number containing four figures, the index is 3. Hence the logarithm of 7438 is (F) To find the logarithm of any whole number, or mixed decimal, to five or six places of figures. RULE. Find the logarithm to the first four figures as above, then take the difference between this logarithm and the next greater in the table. Multiply this difference by the figures you have above four, and cut off so many figures from the right hand of the product as you multiply by, the rest must be added to the logarithm first taken out of the table. Required the logarithm of 59684. The next greater is 77590 Diff. 7 7 diff. 4 remaining figure 2.8 product 3 nearly. Then 77583+3=77586. The natural number consisting of five figures, the index is 4, therefore the logarithm of 59684 (G) To find the logarithm of a pure decimal.. RULE. Find the logarithm corresponding to the significant figures, as if they were whole numbers. Then if the first significant figure be in the place of tenths, the index will be - 1; if in the place of hundredths it will be -2; if in the place of thousandths it will be -3, and so on. Thus, the logarithm of 3754 is •03754 = The logarithm of 003754- 3.57449 The logarithm of 0003754 = 4.57449, &c. PROPOSITION IV. (H) To find the logarithm of a vulgar fraction. RULE. Reduce compound fractions, mixed fractions, &c. to simple fractions. Then subtract the logarithm of the denominator from the logarithm of the numerator; if you carry 1, add it to the index of the logarithm of the denominator; then take the difference of the indices for the index to the logarithm of the fraction. For a proper fraction this index will be-, for an improper fraction it will be +. (I) To find the number answering to any logarithm to four places of figures. RULE. Look in the column of logarithms marked 0 at the top of the table, if you do not find your logarithm there, take the nearest less to it which you can find in that column; the first three figures in the natural number will stand in the column marked No., and looking in the same line from the left hand towards the right, you will either find your logarithm or the nearest less to it in the table, the fourth figure of the natural number will stand at the top of the table. Required the natural number answering to the logarithm 3.90233. Rejecting the index 3, I look in the column 0, where the first three figures of the logarithm are 902, and against 90200 I find the natural number 798, and looking along that line, under 6, I find 90233 the given logarithm, therefore the natural number is 7986, and the index being three, the natural number is a whole number. (K) To find the number answering to any logarithm, to five or six places of figures. |