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1. The numerical calculations which occur in Trigonometry are very much abbreviated by the aid of logarithms. The rules for their use are as follows:

The logarithm of a product is equal to the sum of the logarithms of its factors.

The logarithm of a quotient is equal to the logarithm of the dividend minus the logarithm of the divisor.

The logarithm of any power of a number is equal to the logarithm of the number multiplied by the exponent of the power.

The logarithm of any root of a number is equal to the logarithm of the number divided by the index of the root.

For the investigations of these rules the student is referred to the “Treatise on Trigonometry," p. 87, or to works on Algebra.

The Common Logarithms (Briggs's) are the only ones used in extensive numerical calculations, and the only ones given in the following tables.

The common logarithm of a number is the exponent of that power of 10 which is equal to the number.

Thus, the logarithm of 100 is 2, because 10% = 100. This is usually written log 100 = 2. 10 is the base of the common system.

2. A system of common logarithms means the logarithms of all positive numbers to the base 10.

From the above definition of common logarithms, it follows that 10o = 1, ... log 1 = 0; 10-1=.1,

.. log 0.1 101 = 10,

.. log 10 = 1; 10-2= .01, .. log 0.01 -2, 10% = 100, .. log 100 = 2; 10-3 = .001, .. log 0.001 = -3, 10% = 1000, .. log 1000 = 3; etc.

log 0.0001 = -4, etc.


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Hence, the logarithm of any number between 1 and 10 is some number between 0 and 1; i.e., 0 + a fraction; between 10 and 100 is some number between 1 and 2; i.e., 1 + a fraction, etc., etc.

Thus, it appears that the logarithm of any number greater than 1 is positive, and the logarithm of any positive number less than 1 is negative; and in general the logarithm of a number consists of two parts, an integral part and a decimal part.

The integral part is called the characteristic of the logarithm, and may be either positive or negative.

The decimal part is called the mantissa of the logarithm, and is always kept positive, in order that the mantisse of the logarithms of all numbers expressed by the same digits in the same order may always be the same.

3. It is evident from the above table that the characteristic can always be obtained by the following rule:

The characteristic of the logarithm of a number greater than unity is positive, and one less than the number of digits preceding the decimal point.

The characteristic of the logarithm of a number less than unity is negative, and one more than the number of ciphers immediately after the decimal point.

Thus, the characteristics of the logarithms of 3406, 340.6, 34.06, 3.406, .3406, .0003406, are respectively, 3, 2, 1, 0,-1,-4; the mantissæ are the same, being .53224.

Hence, log .0003406 = 4.53224, the minus sign being written over the characteristic to indicate that it only is negative, the mantissa being always positive.

4. In practice it is more common to avoid the use of negative characteristics by increasing them by 10, and then by allowing for it in the interpretation of the results.

NOTE. — It is only in rare cases that more than seven places of the mantissa are required ; in general, four or five are sufficient; and it is only for the most accurate computations that six or seven are used.

5. A table of logarithms is a table by which the logarithm of any given number, or the number corresponding to any given logaTABLE I. LOGARITHMS OF NUMBERS (Pages 1-19).

6. This table gives the mantissæ of the logarithms of the natural numbers from 1 to 10009, calculated to five decimal places.* The characteristics are determined by the rule in Art. 3. On p. 1, both the characteristic and the mantissa are given.

7. To find the logarithm of a given number.
(1) For a number of one, two, or three figures only.

If the number has one or two figures, find it on page 1 in the column headed N. Then in the same horizontal line as the mumber, and in the next column headed Log, will be found its logarithm.

Thus, log 7 = 0.84510; log 68 = 1.83251.

If the number has three figures, find on one of the pages 2–19, in the column headed N, the given number. Then in the same horizontal line as the number, and in the next vertical column, which is headed 0, will be found the mantissa of its logarithm: prefix the characteristic by the rule in Art. 3.

Thus, log 415 = 2.61805; log 94.8 = 1.97681.

Note 1. A dash under a terminal 5 indicates that the true value is less than 5. Thus the logarithm of 415 to seven decimal places is 2.6180481. If only five decimal places are required, we neglect the 81 and increase 4 to 5. If six decimal places are required, the 1 is neglected; thus the above logarithm is written 2.618048.

(2) For a number of four figures.

Find on one of the pages 2–19, in the column headed N, the first three figures of the given number. Then in the same hori. zontal line as the first three figures, and in the vertical column which has the fourth figure of the given number at the top, will be found the last three figures of the mantissa of the required logarithm, to which the first two figures in the nearest mantissa above, in the column headed 0, are to be prefixed; supply the characteristic by the rule in Art. 3.

Note 2. To save space, only the last three figures of the mantissæ are given in the columns headed 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and the first two at intervals, in the column under L. When the first two figures are not

* With five decimal places the numbers will be correct to the one hundred-thousandth part of a unit, which is near enough for most practical applications.

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