LOGARITHMS.

Logarithms are made to facilitate troublesome calculations in numbers, reducing the labour necessary to arrive at the same results. Multiplication is performed by addition; division, by subtraction; raising of powers and extracting roots, by multiplication and division.

More generally, logarithms are the numeral exponents of ratios; or, they are a series of numbers in arithmetical progression answering to another series of numbers in geometrical progression; thus:

0, 1, 2, 3, 4, &c., are indices of logarithms

1, 10, 100, 1000, 10000, &c., are geometrical progression

It is evident that the same indices serve equally for any geometric series; and, consequently, there may be an endless variety of systems of logarithms to the same common numbers, by only changing the second term, 2, 3, or 10, &c., of the geometric series of whole numbers.

The logarithms most convenient for practice are such as are adapted to a geometric series, increasing in a tenfold proportion as in the above form; and are those which are found at present in most of the common tables on this subject.

The distinguishing mark of this subject of logarithms is, that the index or logarithm of 10 is 1; that of 100 is 2; that of 1000 is 3, &c. And in decimals, the logarithm of 1-1; (placing the negative sine before the index) that of .01 is -2; that of .001 is -3, &c.

Thus the logarithm of 2705 is 3.432167, the logarithm of 10, 1, &c., will be thus:

To find in the table the logarithmic number to any natural number from 1 to 999.

Seek in the left hand column the number, and against it is the logarithm required; thus:

If the number be more than 1000 and less than 10000, seek in the table for the logarithmic decimal of the first three figures 270, as before; then take out the difference between that number and the next below it 1605, multiply this difference by the last figure 5, add this product to the logarithm first found, apply the index 3, it will be the logarithm required.

And add thereto the proper index to the decimal found 3.432167

If the fourth figure be a cypher, the decimal will be the same as the first three figures, to which must be applied the index 3.

TO FIND THE LOGARITHM OF A VULGAR FRACTION, OR A

MIXED NUMBER.

Rule. Reduce the vulgar fraction to a decimal, and find its logarithm as before—

The decimal of .4375; and the logarithm of this decimal is = -1.640978 or, subtract the logarithm of the denominator from the logarithm of the numerator, and the remainder will be the logarithm, which, being that of a decimal fraction, must always have a negative index. If it be a mixed number, reduce it to an improper fraction, and find the difference of the logarithm of the numerator and denominator as before.

To find the natural number to any given logarithm :

This is to be found by the reverse method to the former: find the proposed logarithm in the table and take out the corresponding number, in which the proper number of integers are to be pointed off, viz. 1 more than the index. In finding the number answering to any given logarithm, the index always shows how far the first figures must be removed from the place of units, viz. to the left hand or integers, when the index is affirmative; but to the right hand or decimals, when it is negative.

If the logarithm be exactly found in the table, the natural number is found in the first column, thus:

If the logarithm cannot be exactly found in the table, take the difference of the next greater and the next less, subtracting the one from the other, as also their natural number the one from the other, and the less logarithm from the one proposed, thus:

As the difference of the first tabular logarithm

Is to the difference of their natural number,

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

To their corresponding numerical difference,

Which, being annexed to the least natural number above taken, gives the natural number sought corresponding to the proposed logarithm.

Example.

To find the natural number answering to the given logarithm

Then as 127: 100 :: 81: 64, nearly the numerical difference, which, added to 3409, gives 3409.64, the number required, marking off two integers, because the index of the given logarithm is 3.

Had the index been negative, then 3.532708, its corresponding number, would have been .340964 wholly a decimal.

MULTIPLICATION OF LOGARITHMS.

Rule. Take out the logarithms of the factors from the table, add them together, their sum will be the logarithm required. Then from the table take out the natural number answering to the same for the product sought; observing to add what is to be carried from the decimal part of the logarithm to the affirmative index, or else subtract it from the negative.*

Also adding the indices together when they are of the same kind, both affirmative, or both negative, but subtracting the less from the greater, when the one is affirmative and the other negative, and prefix the sign of the greater to the remainder.

Rule. From the logarithm of the dividend subtract the logarithm of the divisor, and the number answering to the * This rule is also applied to the areas of right lined figures.

remainder will be the quotient required; observing to change the sign of the index of the divisor from affirmative to negative, or from negative to affirmative; then take the sum of the indices, if they be of the same name, or their difference when of different signs, with the sign of the greater for the index to the logarithm of the quotient; and also when 1 is borrowed in the left hand place of the decimal part of the logarithm, add it to the index of the divisor when the index is affirmative, but subtract it when negative, then let the sign of the index arising from hence be changed, and worked with as before.

PROPORTION, OR THE RULE OF THREE.

Rule. Add the logarithms of the second and third terms together, and subtract the logarithm of the first, the remainder will be the logarithm of the fourth term.

Examples.

If 8.2 be to 12 as 74 is to the fourth number, what is that number?

To the logarithm of

Add the logarithm of

108.29 2.034599