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DEFINITIONS AND PRINCIPLES.
1. THE logarithm of a number is the exponent of the power to which it is necessary to raise a fixed number to produce the given number; that is to say, it represents the number of times a fixed number must be multiplied by itself in order to produce any given number.
The fixed number is called the base of the system. In the common system, this base is 10.
It follows from the above, that the logarithm of any power of 10 is equal to the exponent of that power. If, therefore, a number is an exact power of 10, its logarithm is a whole number.
If a number is not an exact power of 10, its logarithm will not be a whole number, but will be made up of an entire part plus a fractional part, which is generally expressed decimally. The entire part of the logarithm is called the characteristic; the decimal part is called the mantissa.
2. The characteristic of the logarithm of a whole number is positive, and numerically 1 less than the number of places of figures in the given number.
Thus, if a number lies between 1 and 10, its logarithm lies between 0 and 1; that is, it is equal to 0 plus a decimal. If a number lies between 10 and 100, its logarithm is equal to 1 plus a decimal; and so on.
3. The characteristic of the logarithm of a decimal fraction is negative, and numerically 1 greater than the number of O's that immediately follow the decimal point. The characteristic alone, in this case, is negative, the man
tissa being always positive. This is indicated by writing the negative sign over the characteristic: thus, 2.380211 is equivalent to 2+.380211. (See last example, p. 8.)
4. The characteristic of the logarithm of a mixed number is the same as that of its entire part. Thus the mixed number 74.103 lies between 10 and 100; hence its logarithm lies between 1 and 2, as does the logarithm of 74.
5. The logarithm of the product of two numbers is equal to the sum of the logarithms of the numbers.
The logarithm of a quotient is equal to the logarithm of the dividend diminished by that 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.
6. The preceding principles enable us to abridge labor in arithmetical calculations, by using simple addition and subtraction instead of multiplication and division.
MANNER OF USING THE TABLES.
TO FIND THE LOGARITHM OF ANY NUMBER. 1. First find the characteristic by rule 2, 3, or 4, given above.
2. Then, if the number be less than 100, look in column N of the table for 10 times or 100 times the amount of it; opposite this multiple, in column 0, will be found the mantissa.
Thus the logarithm of 6 is 0.778151; that of 84 is 1.924279.
3. If the number lie between 100 and 10000, find the first three figures of it in column N; then pass along a horizontal line until you come to the column headed with the fourth figure of the number. At this place will be found the mantissa.
Thus the logarithm of 7200 is 3.857332; that of 8536 is
4. If the number be greater than 10000, place a decimal point after the fourth figure, thus converting the number into a mixed number. Find the mantissa of the entire part by the method last given. Then take from columu D the corresponding tabular difference, multiply this by the decimal part, and add the product to the mantissa just found. The principle employed is that the differences of numbers are proportional to the differences of their logarithms, when these differences are small.
Thus the logarithm of 672887 is 5.827943; that of 43467 is 4,638160.
5. If the number be a decimal, drop the decimal point, thus reducing it to a whole number. Find the mantissa of the logarithm of this number, and it will be the mantissa required.
Thus the logarithm of .0327 is 2.514548; that of 378.024 is 2.577520.
TO FIND THE NUMBER CORRESPONDING TO A GIVEN
LOGARITHM. 6. The rule is the reverse of those just given. Look in the table for the mantissa of the given logarithm. If it cannot be found, take out the next less mantissa, and also the corresponding number, which set aside. Find the difference between the mantissa taken out and that of the given logarithm; annex as many O's as may be necessary, and divide this result by the corresponding number in column D. Annex the quotient to the number set aside, and then point off from the left hand a number of places of figures equal to the characteristic plus 1; the result will be the number required. If the characteristic is negative, the result will be a pure decimal, and the number of O's which immediately follow the decimal point will be one less than the number of units in the characteristic.
Thus the number corresponding to the logarithm 5.233568 is 171225.296; that corresponding to the logarithm 2.233568 is .0171225.
MULTIPLICATION BY MEANS OF LOGARITHMS.
7. Find the logarithms of the factors, and take their sum; then find the number corresponding to the resulting logarithm,