5512, but the characteristic being 4, the number must consist of five places; hence annexing a zero, you have 55120 for the product of 2756 by 20..

4. Divide 756342 by 27867.

100

Both these numbers exceed the numbers in the tables, still we shall be able to find them with great accuracy. First find the logarithm of 756300, which is 5.878694. The difference between this logarithm and that of 756400 is 58. The difference between 756300 and 756400 is 100, and the difference between 756300 and 756342 is 42. Therefore, if 42 = .42 of 58 be added to the logarithm of 756300, it will give the logarithm of 756342 sufficiently exact, 58 x .42 = 24, rejecting the decimals. 5.878694 + 24 = 5.878718. The 58, and consequently the 24, are decimals of the order of the two last places of the logarithm, but this circumstance need not be regarded in taking these parts. It is sufficient to add them to their proper place.

The table generally furnishes means of taking out this logarithm more easily. As the differences do not often vary an unit for considerable distance among the higher numbers, the difference is divided into ten equal parts, (that is, as equal as possible, the nearest number being used, rejecting the decimal parts) and one part is set against 1, two parts against 2, &c. in a column at the right of the table.

In the present case, then, for the 4 of 58,) we look at these parts and for the 2 (for which we must take 11 is, consequently to obtain which is 1, omitting the decimal. thus:

(for which we are to take against it we find 23, and of 58,) we find 11. But we must take of 11 The operation may stand

5.878694

23

1

5.878718

To find the logarithm of 27867, proceed in the same manner, first finding that of 27860, and then adding of the difference, which will be found at the right hand, as above.

log. of quotient 27.141 1.433628

We find the decimal part of this logarithm is between .433610 and .433770, the former of which belongs to the number 2714, and the latter to 2715. Subtract 433610 from 433628, the remainder is 18. Looking in the column of parts, the number next below 18 is 17, which stands against 1 or of the whole difference.

Put this 1 at the right of 2714, which makes 27141. The characteristic 1 shows that the number is between 10 and 100. Therefore the quotient is 27.141. This quotient is correct to three decimal places.

If the table has no column of differences, take the whole difference between .433610 and .433770, which is 160 for a divisor, the 18 for a dividend, annexing one or more zeros. One place must be given to the quotient for each zero.

L. Since a fraction consists of two numbers, one for the numerator and the other for the denominator, the logarithm of a

fraction must consist of two logarithms; and as a fraction expresses the division of the numerator by the denominator, to express this operation on the logarithms, that of the denominator must be represented as to be subtracted from the nume

rator.

The logarithm of 3 is expressed thus :

log. 3-log. 5 = 0.477121 — 0.698970.

The logarithm of a fraction whose numerator is 1, may be

1

expressed by a single logarithm. For is the same as a“.

α

2

If we would express the logarithm of } for example,

10.4771213, consequently

1 10.477121

=10.477121 =}.

That is, the logarithm of is the same as the logarithm of 3, except the sign, which for the fraction is negative. Any fraction may be reduced to the form but the denominator will

1

an

If the subtraction be actually performed, on the expression of this fraction given above, it will be reduced to the logarithm of a fraction of this form.

.

The number corresponding to the logarithm 0.221849 is 1.666 +, but the sign being negative, shows that the number is

1 1.666+

The logarithms of all common fractions may be obtained in either of the above forms, but they are extremely inconvenient in practice. The first on account of its consisting of two logarithms would be useless as well as inconvenient; because though we might find a logarithm corresponding to any fraction, yet in performing operations, a logarithm would never be found in that form when it was required to find its number.

The second form is inconvenient because it is negative, and also because in seeking the number corresponding to the logarithm, a fraction would frequently be found with decimals in the denominator. It would be much better that the whole fraction should be expressed in decimals. If the fraction is used in the decimal form, the logarithms may be used for them almost as easily as for whole numbers.

Suppose it is required to find the logarithm of .5 or 1.

log. 5—log. 10=0.698970 — 1.—— 1+.698970. Suppose it is required to find the logarithm of .05 or T

The logarithms of 10, 100, 1000, &c. always being whole numbers, we have the two parts distinct. The logarithm of .5 is the same as that of 5 except that it has the number 1 joined to it with the sign, which is sufficient to distinguish it, and show it to be a fraction. The logarithm of .05 also is the same, except that 2 is joined to it. That is, the logarithm of the numerator is positive, and that of the denominator negative.

This negative number joined to the positive fractional part, serves as a characteristic, and is a continuation of the principle shown above; thus

The logarithm of a decimal is the same as that of a whole number expressed by the same figures, with the exception of the characteristic, which is negative for the fraction; being 1 when the first figure on the left is tenths, 2 when the first is hundredths, &c. It is convenient to write the sign over the characteristic thus, T, 2, &c. It is not necessary to put the sign+before the fractional part, for this will always be understood to be positive.

In operating upon these numbers, the same rules must be observed as in other cases where numbers are found connected with the signs + and

-.

When the first figure of the fraction is tenths, the characteristic is 1, when the first is hundredths, the characteristic is 2, &c.

The log. of .25 is log. 25 log. 100

= 1.397940-2

21.397940 = 1.397940.

This is the same as the logarithm of 25, except that the characteristic shows that its first figure on the left is 10ths, or one place to the right of units.

In adding the logarithms, there is 1 to carry from the decimal to the units. This one is positive, because the decimal part is so.