Subtract the logarithm of the denominator from the logarithm of the numerator, the remainder is the logarithm of the fraction.

Reduce the mixed number to an improper fraction, and subtract the logarithm of the denominator from the logarithm of the numerator, the remainder is the logarithm required.

The logarithm of a decimal fraction is found exactly in the same manner as a whole number, therefore the fractional part

of a mixed number may be reduced to a decimal, and the logarithm of the integer and fractional part taken out at once; but the index, in this case, is to be regulated by the number of figures in the integral part.

TO FIND THE NUMBER ANSWERING TO ANY GIVEN LOGARITHM.

RULE.

Look for the given logarithm in the different columns of the table, (neglecting the index) until it be found exactly, or the next less then the three first figures of the corresponding natural number will be found opposite to it, in the left hand column, marked Num. and the fourth figure, immediately above it, at the top of the column, which contains the logarithm. If the index of the given logarithm be 3, the four figures thus found are integers: if the index be 2, the three first figures are integers, and the fourth is a decimal: if the index be 1, the two first figures are integers, and the other two are decimals, and so on*.

If the logarithm cannot be found exactly in the table, and more than four figures are required in the natural number, subtract the next less logarithm in the table from the given loga

* The index of the logarithm of a number being a unit less than the number of figures in the number, the number of figures must be one more than the index of the log. corresponding to it.

rithm, to the remainder annex as many ciphers as there are figures required above four in the natural number; which divide by the difference between the next less and the next greater logarithms, (which will be found in the column, marked Diff.,) and the quotient annexed to the four figures, formerly found, will form the natural number required.

EXAMPLE I.

Required the natural number corresponding to the logarithm, 2.734240?

This logarithm is found opposite to 542 and under 3, and the index being 2, the fourth figure is a decimal, therefore, the corresponding number is 542.3.

EXAMPLE II.

Required the natural number corresponding to the logarithm, 5.132780.

The next less logarithm in the table is .132580, answering to the number 1357; the difference between which and the given logarithm is 200, to which two ciphers being annexed, makes 20000; which divided by 320, the difference between the next less and the next greater logarithm, the quotient is 62, which annexed to 1357, makes 135762, the number required.

The use of logarithms is to shorten laborious calculations, such as multiplying and dividing large numbers, raising powers, extracting roots, &c. This valuable property arises from their nature or connection with the decimal scale, which is such, that adding the logarithms corresponding to any two numbers, produces the logarithm of their product; and subtracting their logarithms, leaves the logarithm of their quotient. Hence, addition of logarithms serves the end of multiplication, and subtraction of logarithms, of division, of their corresponding numbers, &c.

TO PERFORM MULTIPLICATION BY LOGARITHMS.;

RULE.

Add the logarithms of the factors, and the sum is the logarithm of their product.

If there be negative and affirmative indices, their difference is to be taken; or rather use the arithmetical complement of the negative indices, and then add them; only, in this case, tens must be rejected from the sum of the indices.

Subtract the logarithm of the divisor from the logarithm of the dividend, the remainder is the logarithm of the quotient.

RULE.

Add the logarithm of the second and third terms together, and subtract the logarithm of the first term from the sum; the remainder is the logarithm of the fourth term, or answer.

EXAMPLE.

If 497 yards of cloth cost £287; what will 389 yards cost?

TO PERFORM INVOLUTION BY LOGARITHMS;

THAT IS, TO RAISE ANY NUMBER TO ANY POWER.

RULE.

Multiply the logarithm of the given number by the index, or exponent of the power; the product is the logarithm of the power required.