RULE II.-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.

NOTE 1.-In the logarithm of a decimal fraction, the characteristic alone is negative, the mantissa being always positive. This fact is indicated by writing the negative sign over the characteristic thus, 2.371465, is equivalent to 2.371465.

NOTE 2. It is to be observed, that the characteristic of the logarithm of a mixed number is the same as that of its entire part. Thus, the characteristic of the logarithm of 725.4275 is the same as the characteristic of the logarithm of 725.

4. A Table of Logarithms is a table by means of which may be found the logarithm corresponding to any number, or the number corresponding to any logarithm.

In the table appended, the mantissas alone are given; the characteristic may be found by one of the rules of Art. 3.

The mantissa of the logarithm of any number is not changed by multiplying or dividing the number by any exact power of 10. Hence, in finding the mantissa of the logarithm of a number, the position of the decimal point may be changed 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 75, is the same as that of 7500.

5. To find the logarithm of a number between 1000 and 10,000. Find the characteristic by the first rule of Art. 3. To determine the mantissa, find in the column headed "N" the left hand three figures of the given number; then pass along the horizontal line in which these figures are found, to the column headed by the fourth figure of the given number, and take out the four figures found there; pass back again to the column

headed "0," and there will be found in this column, either upon the horizontal line of the first three figures or a few lines above it, a number consisting of six figures, the left-hand two figures of which must be prefixed to the four already taken out. Thus,

Log 89793.953228.

66

If, however, any dots are found at the place of the four figures first taken out, or if in returning to the "0" column any dots. are passed, the two figures to be prefixed are the left-hand two of the six figures of the "0" column immediately below. Dots in the number taken out must be replaced by zeros. Thus,

6. To find the logarithm of a number between 1 and 1000.— Find the characteristic by the first rule of Art. 3. To find the mantissa, fill out the given number to four places of figures (or conceive it to be so filled out) by annexing O's (see Art. 4), and find the mantissa corresponding to the resulting number, as in Art. 5. Thus, to find log. of 75: characteristic is 1, by the rule; the mantissa is the same as that corresponding to 7500, i. e., .875061; hence,

In the same way,

Log 75 1.875061.

Log 2 = 0.301030.

7. To find the logarithm of a number greater than 10,000. Find the characteristic by the first rule of Art. 3. To find the mantissa set aside all of the given number except the left-hand four figures, and find the mantissa corresponding to these four, as in Art. 5; multiply the corresponding tabular difference, found in column "D," by the part of the number set aside, and discard as many of the right-hand figures of the product as there are figures in the multiplier, and add the result thus obtained to

the mantissa already found. If the left-hand figure of those discarded is 5, or more, increase the number added by 1.

NOTE. It is to be observed that the tabular difference, found in column "D," is millionths, and not a whole number; and that, therefore, the result to be added "to the mantissa already found" is millionths.

EXAMPLE. To find the logarithm of 672887: the characteristic is 5; set aside 87, and the mantissa corresponding to 6728 is .827886; the corresponding tabular difference is 65, which multiplied by 87, the part of the number set aside, gives 5655; as there are two figures in the multiplier, discard the right-hand two figures of this product, leaving 56; but as the left-hand figure of those discarded is 5, call the result 57 (which is millionths); adding this 57 to the mantissa already found, will give .827943 for the required mantissa; hence,

In the same way,

Log 672887 = 5.827943.

Log 3710053 = 6.569380.

8. To find the logarithm of a decimal.—Find the characteristic by the second rule of Art. 3. To find the mantissa, drop the decimal point and consider the decimal a whole number. Find the mantissa of the logarithm of this number as in preceding articles, and it will be the mantissa required. Thus,

Log .0327 = 2.514548

Log .378024 = 1.577520.

NOTE. To find the logarithm of a mixed number, find the characteristic by Note 2, Art. 3; then drop the decimal point and proceed as above.

9. To find the number corresponding to a given logarithm.-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 any number of 0's, and divide this result by the corresponding number in the column "D." Annex the quotient to the number set aside, and then, if the characteristic is positive, 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, prefix to the figures obtained a number of O's one less than the number of units in the negative characteristic and to the whole prefix a decimal point; the result, a pure decimal, will be the number required.

EXAMPLE.—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 253.

.. The required number is 171225.296.

The number corresponding to the logarithm 2.233568 is .0171225+.

10. Multiplication by Logarithms.-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.

EXAMPLE. Find the continued product of 3.902, 5971.6, . and .0314728.

Here, the 1 carried added to the 3 gives 4, which added to -2 · gives 2 as the characteristic of the logarithm of the product.

11. Division by Logarithms.-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 I, gives 2 for a result. The subtraction, as in this case, is always to be performed in the algebraic sense.

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