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 method 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 1 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 logarithm 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 827943, and we finally have,

log 672887 = 5.827943.

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 "5."

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 reducng the decimal to a whole number. Find the mantissa of the logarithm of this number, and it will be the mantissa required. Thus,

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

14. 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 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 O's which immediately follow the decimal point will be one less than the number of units in the characteristic.

EXAMPLES.

1. Let it be required to find the number corresponding to the logarithm 5.233568.

The next less mantissa in the table is 233504; the corresponding number is 1712, and the tabular difference is

.. The required mumber is 171225.296.

The number corresponding to the logarithm 2.233568 is .0171225.

2. What is the number corresponding to the logarithm 2.785407 ? Ans. .06101084.

3. What is the number corresponding to the logarithm 1.846741 ? Ans. .702653.

MULTIPLICATION BY MEANS OF LOGARITHMS.

15. From the principle proved in Art. 5, we deduce the following

RULE.

Find the logarithms of the factors, and take their sum, then find the number corresponding to the resulting logarithm, and it will be the product required.

2. Find the continued product of 3.902, 597.16, aud 0.0314728.

Here, the 2 cancels the +2, and the 1 carried from the decimal part is set down.

3. Find the continued product of 3.586, 2.1046, 0.8372, and 0.0294. Ans. 0.1857615.

DIVISION BY MEANS OF LOGARITHMS.

16. From the principle proved in Art. 6, we have the following

RULE.

Find the logarithms of the dividend and divisor, and subtract the latter from the former; then find the number corresponding to the resulting logarithm, and it will be the quotient required.

Here, 1 taken from ī, gives 2 for a result. The subtraction, as in this case, is always to be performed in the algebraic sense.

3. Divide 37.149 by 523.76.

Ans. 0.0709274.

The operation of division, particularly when combined with that of multiplication, can often be simplified by using the principle of

17. The ARITHMETICAL COMPLEMENT of a logarithm is the result obtained by subtracting it from 10.

Thus, 8.130456

The arithmetical

is the arithmetical complement of 1.869544. complement of a logarithm may be written out by commencing at the left hand and subtracting each figure from 9,