6. Dividing (4) by (5), member by member, we have,

That is, the logarithm of a quotient is equal to the logarithm of the dividend diminished by that of the divisor.

7. Raising both members of (4) to the power denoted

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

8. Extracting the root, indicated by r, of both members of (4), we have,

That is, the logarithm of any root of a number is equal to the logarithm of the number divided by the index of the

root.

The preceding principles enable us to abbreviate the oper ations of multiplication and division, by converting them into the simpler ones of addition and subtraction.

TABLE OF LOGARITHMS.

9. A TABLE OF LOGARITHMS, is a table containing a set of numbers and their logarithms, so arranged, that having given any one of the numbers, we can find its logarithm; or, having the logarithm, we can find the corresponding

number.

In the table appended, the complete logarithm is given for all numbers from 1 up to 10,000. For other numbers, the mantissas alone are given; the characteristic may be found by one of the rules of Art. 4.

Before explaining the use of the table, it is to be shown that the mantissa of the logarithm of any number is not changed by multiplying or dividing the number by any exact

power of 10.

Let n

10o any

represent any number whatever, and 10" p being any whole number, either positive or negative. Then, in accordance with the principles of Arts.

power of 10,

5 and 3, we shall have,

but

p is, by hypothesis, a whole number: hence, the deci mal part of the log (n × 10") is the same as that of log n ; which was to be proved.

Hence, in finding the mantissa of the logarithm of a num ber, we may regard the number as a decimal, and move the decimal point to the right or left, at pleasure. Thus, the mantissa of the logarithm of 456357, is the same as that of the number 4563.57; and the mantissa of the logarithm of 2.00357, is the same as that of 2003.57.

1o.

10.

MANNER OF USING THE TABLE.

To find the logarithm of a number less than 100.

Look on the first page, in the column headed "N,"

for the given number; the number opposite is the logarithm required.

Thus,

2o.

To find the logarithm of a number between 100 and

10,000.

11. Find the characteristic by the first rule of Art. 4. To find the mantissa, look in the column headed "N," for the first three figures of the number; 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 four figures of the mantissa, to which, two other figures, taken from the column headed "0," are to be prefixed. If the figures found stand opposite a row of six figures, in the column headed "0," the first two of this row are the ones to be prefixed; if not, ascend the column till a row of six figures is found; the first two, of this row, are the ones to be prefixed.

If, however, in passing back from the four figures, first found, any dots are passed, the two figures to be prefixed must be taken from the line immediately below. If the figures first found fall at a place where dots occur, the dots must be replaced by 0's, and the figures to be prefixed must be taken from the line below. Thus,

3. To find the logarithm of a number greater than 10,000.

12. Find the characteristic by the first rule of Art. 4. To find the mantissa, place a decimal point after the fourth figure (Art. 9), thus converting the number into a mixed number. Find the mantissa of the entire part, by the me thod last given. Then take from the column headed "D," the corresponding tabular difference, and multiply this by the decimal part and add the product to the mantissa just found. The result will be the required mantissa.

It is to be observed that when the decimal part of the product just spoken of is equal to or exceeds 5, we add to the entire part, otherwise the decimal part is rejected.

EXAMPLE.

1. To find the logarithm of 672887.

The characteristic is 5. Placing a decimal point after the fourth figure, the number becomes 6728.87. The mantissa of the Togarithm of 6728 is 827886, and the corresponding number in the column "D" is 65. Multiplying 65 by .87, we have 56.55; or, since the decimal part exceeds .5, 57. We add 57 to the mantissa already found, giving and we finally have,

log 672887 => 5.827943.

827943,

The numbers in the column "D" are the differences between the logarithms of two consecutive whole numbers, and are found by subtracting the number under the heading "4" from that under the heading 65."

In the example last given, the mantissa of the logarithm of 6728 is 827886, and that of 6729 is 827951, and their difference is 65; 87 hundredths of this difference is

57 hence, the mantissa of the logarithm of 6728.87 is found by adding 57 to 827886. The principle employed is, that the differences of numbers are proportional to the differences of their logarithms, when these differences are small.

4°. To find the logarithm of a decimal.

13. Find the characteristic by the second rule of Art. 4. To find the mantissa, drop the decimal point, thus reduc ing the decimal to a whole number. Find the mantissa of the logarithm of this number, and it will be the mantissa required. Thus,

log .0327 = 2.514548
log 378.024 = 2.577520

5°. To find the number corresponding to a given logarithm.

Look

If it

14. The rule is the reverse of those just given. in the table for the mantissa of the given logarithm. cannot be found, take out the next less mantissa, and also the corresponding number, which set aside. Find the differ ence 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 the 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 characterististic 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 0's which immediately follow the decimal point will be one less than the number of units in the characteristic.