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Hence it appears that the logarithm of every number between 1 and 0.1 is some number between 0 and -1, or may be

rep resented by - 1 plus a fraction; the logarithm of every number between 0.1 and .01 is some number between – 1 and -2, or may be represented by —2 plus a fraction; the logarithm of every number between .01 and .001 is some number between – 2 and —3, or is equal to -3 plus a fraction, and

so on.

The logarithms of most numbers, therefore, consist of an integer and a fraction. The integral part is called the characteristic, and may be known from the following

Rule. The characteristic of the logarithm of any number greater than unity, is one less than the number of integral figures in the given number.

Thus the logarithm of 297 is 2 plus a fraction; that is, the characteristic of the logarithm of 297 is 2, which is one less than the number of integral figures. The characteristic of the logarithm of 5673.29 is 3; that of 73254.1 is 4, &c.

The characteristic of the logarithm of a decimal fraction is a negative number, and is equal to the number of places by which its first significant figure is removed from the place of units.

Thus the logarithm of .0046 is - 3 plus a fraotion; that is, the characteristic of the logarithm is – 3, the first significant figure, 4, being removed three places from units.

(3.) Since powers of the same quantity are multiplied by adding their exponents (Alg., Art. 50),

The logarithm of the product of two or more factors is equal to the sum of the logarithms of those factors.

Hence we see that if it is required to multiply two or more numbers by each other, we have only to add their logarithms: the sum will be the logarithm of their product. We then look in the table for the number answering to that logarithm, in order to obtain the required product.

Also, since powers of the same quantity are divided by subtracting their exponents (Alg., Art. 66),

The logarithm of the quotient of one number divided by an.

other, is equal to the difference of the logarithms of those numbers.

Hence we see that if we wish to divide one number by another, we have only to subtract the logarithm of the divisor from that of the dividend; the difference will be the logarithm of their quotient.

(4.) Since, in Briggs' system, the logarithm of 10 is 1, if any number be multiplied or divided by 10, its logarithm will be increased or diminished by 1; and as this is an integer, it will only change the characteristic of the logarithm, without affecting the decimal part. Hence

The decimal part of the logarithm of any number is the same as that of the number multiplied or divided by 10, 100, 1000, &c. Thus, the logarithm of 65430

is 4.815777; 6543

is 3.815777; 654.3 is 2.815777; 65.43 is 1.815777 6.543

is 0.815777; .6543 is 7.815777; .06543 is 2.815777;

.006543 is 3.815777 The minus sign is here placed over the characteristic, to show that that alone is negative, while the decimal part of the logarithm is positive.

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TABLE OF LOGARITHMS.

(5.) A table of logarithms usually contains the logarithms of the entire series of natural numbers from 1 up to 10,000, and the larger tables extend to 100,000 or more. In the smaller tables the logarithms are usually given to five or six decimal places; the larger tables extend to seven, and sometimes eight or more places.

In the accompanying table, the logarithms of the first 100 numbers are given with their characteristics; but, for all other numbers, the decimal part only of the logarithm is given, while the characteristic is left to be supplied, according to the rule in Art. 2.

(6.) To find the Logarithm of any Number between 1 and 100.

Look on the first page of the accompanying table, along the column of numbers under N., for the given number, and against it, in the next column, will be found the logarithm with its characteristic. Thus, opposite 13 is 1.113943, which is the logarithm of 13; 65 is 1.812913,

65.

To find the Logarithm of any Number consisting of three

Figures. Look on one of the pages of the table from 2 to 20, along the left-hand column, marked N., for the given number, and against it, in the column headed 0, will be found the decimal part of its logarithm. To this the characteristic must be prefixed, according to the rule in Art. 2. Thus the logarithm of 347 will be found, from page 8, 2.540329; 871

18, 2.940018. As the first two figures of the decimal are the same for several successive numbers in the table, they are not repeated for each logarithm separately, but are left to be supplied. Thus the decimal part of the logarithm of 339 is .530200. The first two figures of the decimal remain the same up to 347; they are therefore omitted in the table, and are to be supplied.

To find the Logarithm of any Number consisting of four

Figures. Find the three left-hand figures in the column marked N., as before, and the fourth figure at the head of one of the other columns. Opposite to the first three figures, and in the column under the fourth figure, will be found four figures of the logarithm, to which two figures from the column headed 0 are to be prefixed, as in the former case. The characteristic must be supplied according to Art. 2. Thus the logarithm of 3456 is 3.538574;

8765 is 3.942752. In several of the columns headed 1, 2, 3, &c., small dots are found in the place of figures. This is to show that the two figures which are to be prefixed from the first column have changed, and they are to be taken from the horizontal line di

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rectly below. The place of the dots is to be supplied with ciphers. Thus the logarithm of 2045 is 3.310693;

9777 is 3.990206. The two leading figures from the column 0 must also be taken from the horizontal line below, if any dots have been passed over on the same horizontal line. Thus

the logarithm of 1628 is 3.211654.

To find the Logarithm of any Number containing more than

four Figures. (7.) By inspecting the table, we shall find that, within certain limits, the differences of the logarithms are nearly proportional to the differences of their corresponding numbers. Thus the logarithm of 7250 is 3.860338;

7251 is 3.860398;
7252 is 3.860458;

7253 is 3.860518. Here the difference between the successive logarithms, called the tabular difference, is constantly 60, corresponding to a difference of unity in the natural numbers. If, then, we suppose the logarithms to be proportional to their corresponding numbers (as they are nearly), a difference of 0.1 in the numbers should correspond to a difference of 6 in the logarithms; a difference of 0.2 in the numbers should correspond to a difference of 12 in the logarithms, &c. Hence the logarithm of 7250.1 must be 3.860344; 7250.2

3.860350; 7250.3

3.860356. In order to facilitate the computation, the tabular difference is inserted on page 16 in the column headed D., and the proportional part for the fifth figure of the natural number is given at the bottom of the page. Thus, when the tabular difference is 60, the corrections for .1, 2, .3, &c., are seen to be 6, 12,

18, &c.

If the given number was 72501, the characteristic of its logarithm would be 4, but the decimal part would be the same as for 7250.1.

If it were required to find the correction for a sixth figure

in the natural number, it is readily obtained from the Proportional Parts in the table. The correction for a figure in the sixth place must be one tenth of the correction for the same figure if it stood in the fifth place. Thus, if the correction for .5 is 30, the correction for .05 is obviously 3.

As the differences change rapidly in the first part of the table, it was found inconvenient to give the proportional parts for each tabular difference; accordingly, for the first seven pages, they are only given for the even differences, but the proportional parts for the odd differences will be readily found by inspection.

Required the logarithm of 452789.
The logarithm of 452700 is 5.655810.

. The tabular difference is 96.

Accordingly, the correction for the fifth figure, 8, is 77, and for the sixth figure, 9, is 8.6, or 9 nearly. Adding these corrections to the number before found, we obtain 5.655896.

The preceding logarithms do not pretend to be perfectly exact, but only the nearest numbers limited to six decimal places. Accordingly, when the fraction which is omittid exceeds half a unit in the sixth decimal place, the last figure must be increased by unity. Required the logarithm of 8765432.

The logarithm of 8765000 is 6.942752
Correction for the fifth figure, 4,

20
sixth figure, 3,

1.5
seventh figure, 2,

0.1
Therefore the logarithm of 8765432 is 6.942774.
Required the logarithm of 234567.
The logarithm of 234500 is

5.370143
Correction for the fifth figure, 6,

111 sixth figure, 7,

13 Therefore the logarithm of 234567 is

5.370267

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To find the Logarithm of a Decimal Fraction (8.) According to Art. 4, the decimal part of the logarithm of any number is the same as that of the number multiplied or divided by 10, 100, 1000, &c. Hence, for a decimal frac.

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