II. To find, in the Table, the natural number to any
Logarithm.

This is to be found by the reverse method to the former, namely, by searching for the proposed logarithm among those in the table, and taking out the corresponding number by inspection, in which the proper number of integers is to be pointed off, viz. 1 more than the units of the affirmative index. For, in finding the number answering to any given logarithm, the index always shows how far the first figure must be removed from the place of units to the left or in integers, when the index is affirmative; but to the right or in decimals, when it is negative.

EXAMPLES.

So, the number to the logarithm 1'5326525 is 34 092. And the number of the logarithm—1'5326525 is 34092. But if the logarithm cannot be exactly found in the table, take out the next greater and the next less, subtracting one of these logarithms from the other, and also one of their natural numbers from the other, and the less logarithm from the logarithm proposed. Theri say,

As the first difference, or that of the tabular logarithms, Is to the difference of their natural numbers,

So is the difference of the given logarithm and the last tabular logarithm

To their corresponding numeral difference.

Which being annexed to the least natural number above taken, the natural number corresponding to the proposed logarithm is obtained.

EXAMPLE.

Find the natural number answering to the given logarithm

1*5326606.

V«|. I.

Ee

Here the next greater and next less tabular logarithms, with their corresponding numbers, &c. are as below.

Next greater 5326652 its num. 34O9300; giv.log. 532660S Next less 5326525 its num. 3409200; next less 5326525

Differenc es

127

100

81

Then, as 127: 100 :: 81: 64 nearly, the numeral differ

ence.

Therefore 34'09264 is the number sought, two integers being marked off, because the index of the given logarithm is 1. Had the index been negative, thus, —1*5326606, its corresponding number would have been 3409264, wholly decimal.

Or, the proportional numeral difference may be found, in the best tables, by inspection of the small tables of proportional parts, placed in the margin.

MULTIPLICATION BY LOGAf HMS.

RULE.

Take out the logarithms of the factors from the table then add them together, and their sum will be the logarithm of the product required. Then, by means of the table, take out the natural number answeripg to the sum, for the pro* duct sought.

Note 1. In every operation, what is carried from the decimal part of a logarithm to its index is affirmative; and is therefore to be added to the index, when it is affirmative; but subtracted, when it is negative.

Note 2. When the indices have like signs, that is, both or both, they are to be added, and the sum has the common sign; but when they have unlike signs, that is, one + and the other, their difference, with the sign of the greater, is to be taken for the index of the sum.

3. To multiply 3902, 597'16, and '0314728 all together.

Here the 2 cancels the 2, and the 1, to be carried from the decimals, is set down.

4. To multiply 3'586, 2:1046, 0'8372, and 0'0294 all together.

Here the 2, to be carried, cancels the-2, and there re

mains the -1 to be set down,

DIVISION BY LOGARITHMS.

RULE

From the logarithm of the dividend subtract the logarithm of the divisor, and the number answering to the remainder will be the quotient required.

Note. If 1 be to be carried to the index of the suhtrahend, apply it according to the sign of the index; then change the sign of the index to —, if it be +, or to +, if it be; and proceed according to the second note under the last rule,

Here 1, carried from the decimals to the-3, makes it be

come-2, which, taken from the other 2, leaves 0 remaining.

4. To divide 7438 by 12'9476,

Here the 1, taksa from the 1, makes it become be set down.

INVOLUTION BY LOGARITHMS.

RULE.

Multiply the logarithm of the given number by the index of the power, and the number answering to the product will be the power required.

Note. A negative index, multiplied by an affirmative number, gives a negative product; and as the number, carried from the decimal part, is affirmative, their difference with the sign of the greater is, in that case, the index of the product.