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numbers; from the above two logarithms, and the logarithm of 10, which is 1, we may raise a great many logarithms, as in the following examples :
EXAMPLE 6. Because 2 x 2 = 4, therefore Because 32 = 9, therefore to log. 2
•301029995 log. 3 • 47712125416 add log. 2 •3010299953 mult. by 2
sum is log. 4 •602059991 gives log. I
EXAMPLE 7. Because 2 X 3 6, therefore Beeause , 5, therefore to log. 2
•301029995 from log. 10 1.000000000 add log. 3. - •477121255 take log. 2 *301029995
sum is log. 6
leaves log. 5
EXAMPLE 8. Because 2 = 8, therefore Because 3 x 4 = 12, therefore log. 2
•301029995} to log. 3 •477121255 mult, by 3
gives log. 8
gives log. 12
And thus, computing, by this general rule, the logarithms to the other prime numbers, 7, 11, 13, 17, 19, 23, &c. and then using composition and division, we may easily find as many logarithms as we please, or may speedily examine any logarithm in the table*.
* There are, besides these, many other ingenious methods, which later writers have discovered for finding the logarithms of numbers, in a much easier way than by the original inventor ; but, as they cannot be understood without a knowledge of some of the higher branches of the mathematics, it is thought proper to omit them, and to refer the reader to those works which are written expressly on the subject. It would likewise much exceed the limits of this compendium, to point out all the particular artifices that are made use of for constructing an entire table of these numbers ; but any information of this kind, which the learner may wish to obtain, may be found in my Tables, before mentioned.
Description and Use of the TABLE of LOGARITHMS.
HAVING explained the manner of forming a table of the logarithms of numbers, greater than unity ; the next thing to be done is, to show how the logarithms of fractional quantities may be found. In order to this, it may be observed, that as in the former case a geometric series is supposed to increase towards the left, from unity, so in the latter case it is supposed to decrease towards the right hand, still beginning with unit; as exhibited in the general description, page 148, where the indices being made negative, still show the logarithms to which they belong. Whence it appears, that as + 1 is the log. of 10, so i is the log. of to or:1; and as + 2 is the log. of 100, so 2 is the log. of jóo or 01: : and so on.
Hence it appears in general, that all numbers which consist of the same figures, whether they be integral, or frace tional, or mixed, will have the decimal parts of their logarithms the same, but differing only in the index, which will be more or less, and positive or negative, according to the place of the first figure of the number.
Thus, the logarithm of 2651 being 3.423410, the log. of to, or to or Todo, &c. part of it ; will be as follows: Numbers.
Logarithms. 2 0 5 1
3 4 2 3 4 1 0 2 6 5.1
2 4 2 3 4 1 0 2 6.5 1
1 4 2 3 4 10 2 6 5 1
0.4.2 3 4 1 0 •2 6 5 1
1 4 2 3 4 1 0 •0 2 6 5 1
- 2 4 2 3 4 1 0 • 0 0 2 6 5 1
-3 -4 2 3 4 1 0 Hence it also appears, that the index of any logarithm, is always less by 1 than the number of integer figures which the natural number consists of ; or it is equal to the distance of the first figure from the place of units, or first place of integers, whether on the left, or on the right, of it : and this index is constantly to be placed on the left-hand side of the decimal part of the logarithm.
When there are integers in the given number, the index is always affirmative ; but when there are no integers, the index is negative, and is to be marked by a short line drawn before it, or else above it. Thus,
A number having 1, 2, 3, 4, 5, &c. integer places,
And a decimal fraction having its first figure in the
1st, 2d, 3d, 4th &c. place of the decimals, has alwags -1, -2, -3, -*4. &c. for the index of its logurithm. It may also be observed, that though the indices of frac tional quantities are negative, yet the decimal pails of thei 1 garithms are always affirmative. And the negative mark (may be set either before the index or over it.
1. TO FIND, IN THE TABLE, THE LOGARITHM TO ANY
1. If the given Number be less than 100, or consist of only two figures; its log. is immediately found by inspection in ihe first page of the table, which contains all numbers from 1 to 100, with their logs. and the index imniediately annexed in the next column.
So the log. of 5 is 0 698970. The log. of 23 is 1:361728. The log. of 50 is 1 698970
And so on. 2. If the Number be more than 100 but less than 10000; that is, consisting of either three or four figures; the decimal part of the logarithm is found by inspection in the othe pages of the table, standing against the given number, in this manner ; viz. the first three figures of the given number in the first column of the page, and the fourth figure one of those along the top line of it ; then in the angle of mecting are the last four figures of the logarithm, and the first two figures of the same at the beginning of the same line in the second column of the page : to which is to be prefixed the proper index, which is always 1 less than the number of integer figures.
So the logarithm of 251 is 2.39967 4, that is, the decimal •399674 found in the table, with the index 2 prefixed, because the given number contains 'three integers. And the log of 34 09 is 1.532627, that is, the decimal •532627 found in the table, with the index I prefixed, because the given number contains two integers.
3. But if the given Number contain more than four figures ; take out the logarithm of the first four figures by inspection in the table, as before, as also the next greater logarithm, subtracting the one logarithm from the other, as also their corresponding numbers the one from the other. Then say,
As the difference between the two numbers,
See the table of Logarithms at the end of the 2d volume.
Which part being added to the less logarithm, before taken out, gives the whole logarithm sought very nearly.
To find the logarithm of the number 34:0926.
Then as 100 : 127 :: 26 : 33, the proportional part.
532627, the first log.
4. If the number consist both of integers and fractions, or is entirely fractional ; find the decimal part of the logarithm the same as if all its figures were integral; then this, having prefixed to it the proper index, will give the logarithm required.
5. And if the given number be a proper vulgar fraction : subiract the logarithm of the denominator from the logarithm of the numerator, and the remainder will be the logarithm sought ; which, being that of a decimal fraction, must always have a negative index.
6. But if it be a mixed number; reduce it to an improper fraction, and find the difference of the logarithms of the numerator and denominator, in the same manner as before.
Log. of 37
1. To find the log. of 1. 2. To find the log. of 175.
1.568202 First, 1743 = . Then, 1 973128 Log..of 405
1.361728 Dif. log. of 31 - 1.595074
Dif. log. of 1711 1.245727 Where the index 1 is negative.
Log. of 23
II. TO FIND THE NATURAL NUMBER TO ANY GIVEN
This is to be found in the tables. by the reverse method to the former, namely, by searching for the proposed loga. rithm among those in the table, and taking out the corresponding number by inspection, in which the proper number of integers are to be pointed off, viz. I more than the index. For, in finding the number answering to any given logarithm, the index always shows how far the first figure
must be removed from the place of units, viz. to the left hand, or integers, when the index is affirmative ; but to the right hand, or decimals, when it is negative.
So, the number to the log. 1.532882 is 34:11.
And the number of the log. 1.532882 is •3411.
But if the logarithm cannot be exactly found in the table; take out the next greater and the next less, subtracting the one of these logarithms from the other, as also their natural numbers the one from the other, and the less logarithm from the logarithm proposed. Then say,
As the difference of the first or tabular logarithms,
To their corresponding numeral difference. Which being annexed to the least natural number above taken, gives the natural number sought, corresponding to the proposed logarithm.
So, to find the natural number answering to the given
logarithm 1:532708, Here the next greater and next less tabular logarithms, with their corresponding numbers, are as below:
Next greater 532744 ils num. 341000; given log. 532708 Next less 532627 its num. 340900 ; next less 532627
Then, as 127 : 100 :: 81 : 64 nearly, the numeral differ, Therefore 34.0964 is the number sought, marking off two integers, because the index of the given logarithm is 1.
Had the index been negative, thus 1.532708, its corresponding number would have been •340965, wholly de. cimal.