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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
fractions, thus if yn denote any number, then will 1 .gn .
yn, &c. constitute an increasing series of numbers in geometrical progression, of which the indices
&c. are the logarithms, or
101 - I.co
will constitute a decreasing
(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 10'; the logarithm of 100 is 2, being the index of 102; the logarithm of 1000 is 3, being the index of 10%; the logarithm of 10000 is 4, being the index of 10^, &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.
Again, the logarithm of so riði is, 1, being the index of 10 ; the logarithm of is —2, being the index of 10
; the logarithm of 1000 = ios is—3, being the index of 10 &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 10x = a, 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.
If 103* =2, then x=0•30103 the log. of 2.
The logarithm of 94261 = .97433 = x, viz. 107 = 10 =9.4261, multiply by 10.
=94.261. Multiply again by 10,
2C + 2 2.97433 And 10
=10 =942:61. Multiply a third time by 10, 2 + 3
S.97433 Then 10
=10 =9426•1. Multiply once more by 10, * +4
4.97433 Then 10
=10 =94261 and so on. If instead of multipying by 10, we divide by 10 in the equation 10x=9.4261, then will -I
•97433 -1 -1.97433 10 =10
= .94261 ;
The design of this chapter is to show the nature and properties of logarithms, and not the different methods of constructing them.
Divide again by. 10,
-1.97433-1 Then 10°
= .0094261, &c. 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.
THE USE OF THE TABLE OF LOGARITHMS.
(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
Log. No. Log. Examp. 37=1.56820 3754 = 3:57449 4980=3•69723
4=0*60206 375.42.57449 7986 =3.90233 91.6=1.96190 37.54=1:57449 3700=3•56820 9.16=0.96190 3.754 = 0.57449 4000=3•60206
(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.
Diff. 7 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 is 4077586. Examples. No. Log. No. Log.
59.684 = 1•77586 65.439 = 1.81584
94268 = 4.97436 27863 = 4-44503 785920 = 5.89538 131755 511977
(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 1:57449
PROPOSITION IV. (H) To find the logarithm of a vulgar fraction.
RULE. Reduce compound fractions, mixed fractions, &c.
The logarithm of 5 = 0.69897
The logarithm of '= - 1.44370 In the same manner the logarithm of 3 = - 1.99535; the log. of = -1•88303; the log. of } = -1•77815; the log. of $349 = + 0.72870, &c.
PROPOSITION V. (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
1•76343 = 58 2.85775 = 72007
PROPOSITION VI. (K) To find the number answering to any logarithm, to five or six places of figures.