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That is, all numbers which are tenfold, the one of the other, have the same logarithm.
376200 has for its logarithm - 5.575419.
16 - 0.575419. When a number consists of whole. numbers and decimal parts, we find the fractional part of the logarithm in the same manner as if all the figures of the number belonged to the whole number, but we give it the index corresponding to the whole number only.
In most tables of logarithms they are carried as far as seven decimal places. Some however are only carried to five or six. The disposition of the tables is something different in different sets, but they are generally accompanied with an explanacion. When one set of tables is well understood, all others will be easily learned. The logarithms for the following examples may be found in any table of logarithms. They are used here as far as six places..
1. Multiply 43 by 25.
Find 43 in the column of numbers, and against it in the column of logarithms you will find 1.633468, and against 25 you will find 1.397940. Add these two logarithms together and their sum is the logarithm of the product. log. 43 . i . . 1.633468
25 . . . 1.397940
6 1075 . . . . 3.031408 Find this logarithm in the column of logarithms, and against it in the column of numbers you find 1075 which is the product of 43 multiplied by 25. The index, 3, shows that the number must consist of four places.
Let the learner prove the results at first by actual multiplication.
2. Multiply 2520 by 300.
By what was remarked above, the logarithm of 2520 is the - same as that of 252 with the exception of the index, and that of 300 is the same as that of 3 except the index. .
Find the number 252 in the left hand column, and against it in the second column you find .401401. The number 2520 consists of four places, therefore the index of its logarithm'must be (4-1) or 3. The logarithm corresponding to 300 is .477121, and its index must be 2, because 300 consists of three places. log. 2520 .. .
. 3.401401 300. . . . 2.477121
56 756000 . . . 5.878522 Find this logarithm, and against it in the column of numbers you will find 756 ; but the index 5 shows that the number must consist of 6 places; therefore three zeros must be annexed to the right, which makes the number 756000, which is the product of 2520 by 300.
3. Multiply 2756 by 20.
* To find the logarithm of 2756, find in the column of numbers 275, and at the top of the table look for 6. In the column under 6 and opposite 275 you find .440279 for the decimal part of the logarithm of 2756. The characteristic will be 3.
log. 2756 ; . . 3.440279
6 55120 . . . . 4.741309 Looking in the table for this logarithm, against 551 you will find .741152 and against 552 you will find .741939, The logarithm .741309 is between these two. Against 551, look along in the other columns. In the column under 2 you find the logarithm required. The figures of the number, then, are
* In some tables the whole number 2756 may be found in the left hand column.
5512, but the characteristic being 4, the number must consist of five places; hence annexing a zero, you have 55120 for the product of 2756 by 20.
4. Divide 756342 by 27867.
Both these numbers exceed the numbers in the tables, still we shall be able to find them with great accuracy. First find the logarithm of 756300, which is 5,878694. The difference between this logarithm and that of 756400 is 59. The difference between 756300 and 756400 is 100, and the difference between 756300 and 756342 is 42. Therefore, if 43 = .42 of 58 be added to the logarithm of 756300, it will give the logarithm of 756342 sufficiently exact, 53 x .42 = 24, rejecting the decimals. 5.878694 + 24 = 5.878713. The 58, and consequently the 24, are decimals of the order of the two last places of the logarithm, but this circumstance need not be regarded in taking these parts. It is sufficient to add them to their proper place.
The table generally furnishes means of taking out this logarithm more easily. As the differences do not often vary an unit for considerable distance among the higher numbers, the difference is divided into ten equal parts, (that is, as equal as possible, the nearest number being used, rejecting the decimal parts) and one part is set against 1, two parts against 2, &c. in a column at the right of the table. . .
In the present case, then, for the 4 (for which we are to take to of 58, we look at these parts and against it we find 23, and for the 2 (for which we must take és of 58,) we find 11. But. 11 is o, consequently to obtain jão we must take is of 11 which is 1, omitting the decimal. The operation may stand thus :
log. 756300 5.878694
log. 756342 5.878718 To find the logarithm of 27867, proceed in the same manner, first finding that of 27860, and then adding of the difference which will be found at the right hand, as above.
log. of quotient 27.141 1.433628 .. We find the decimal part of this logarithm is between
.433610 and .433770, the former of which belongs to the number 2714, and the latter to 2715. Subtract 433610 from 433628, the remainder is 18. Looking in the column of parts, the number next below 18 is 17, which stands against 1 or to of the whole difference.
Put this 1 at the right of 2714, which makes 27141. The characteristic 1 shows that the number is between 10 and 100. Therefore the quotient is 27.141. This quotient is correct to three decimal places.
If the table has no column of differences, take the whole difference between .433610 and .433770, which is 160 for a divisor, the 18 for a dividend, annexing one or more zeros. One place must be given to the quotient for each žero.
L. Since a fraction consists of two numbers, one for the nunierator and the other for the denominator, the logarithm of a
fraction must consist of two logarithms ; and as a fraction expresses the division of the numerator by the denominator, to express this operation on the logarithms, that of the denominator must be represented as to be subtracted from the nume'rator. The logarithm of is expressed thus :
log. 3 - log. 5 = 0.477121 — 0.698970. The logarithm of a fraction whose numerator is 1, may be expressed by a single logarithm. For is the same as a If we would express the logarithm of } for example,
10-41719 = 3, consequently ...11 = 10^-719 = $. That is, the logarithm of 4 is the same as the logarithm of 3, except the sign, which for the fraction is negative. Any fraction may be reduced to the form -, but the denominator will consist of decimals or still contain a fraction.
=i = 1.666 F If the subtraction be actually performed, on the expression of this fraction given above, it will be reduced to the logarithm of a fraction of this form.
0.477121 -0.698970 = -0.221849. The number corresponding to the logarithm 0.221849 is 1.666 +, but the sign being negative, shows that the number is
1.666 + The logarithms of all common fractions may be obtained in either of the above forms, but they are extremely inconvenient in practice. The first on account of its consisting of two logarithms would be useless as well as inconvenient; because though we might find a logarithm corresponding to any fraction, yet in performing operations, a logarithm would never be found in that form when it was required to find its nuinber,