tion, we find the logarithm as if the figures were integers, and prefix the characteristic according to the rule of Art. 2. To find the Logarithm of a Vulgar Fraction. (9.) We may reduce the vulgar fraction to a decimal, and find its logarithm by the preceding article; or, since the value of a fraction is equal to the quotient of the numerator divided by the denominator, we may, according to Art. 3, subtract the logarithm of the denominator from that of the numerator; the difference will be the logarithm of the fraction. Ex. 1. Find the logarithm of, or 0.1875. From the logarithm of 3, 0.477121, Take the logarithm of 16, 1.204120. Leaves the logarithm of, or .1875, 1.273001. To find the Natural Number corresponding to any Logarithm. (10.) Look in the table, in the column headed 0, for the first two figures of the logarithm, neglecting the characteristic; the other four figures are to be looked for in the same column, or in one of the nine following columns; and if they are exactly found, the first three figures of the corresponding number will be found opposite to them in the column headed N., and the fourth figure will be found at the top of the page. This number must be made to correspond with the characteristic of the given logarithm by pointing off decimals or annexing ciphers. Thus the natural number belonging to the log. 4.370143 is 23450; 1.538574 is 34.56. 66 If the decimal part of the logarithm can not be exactly found in the table, look for the nearest less logarithm, and take out the four figures of the corresponding natural number as before; the additional figures may be obtained by means of the Proportional Parts at the bottom of the page. Required the number belonging to the logarithm 4.368399. On page 6, we find the next less logarithm .368287. The four corresponding figures of the natural number are 2335. Their logarithm is less than the one proposed by 112. The tabular difference is 186; and, by referring to the bottom of page 6, we find that, with a difference of 186, the figure corresponding to the proportional part 112 is 6. Hence the five figures of the natural number are 23356; and, since the characteristic of the proposed logarithm is 4, these five figures are all integral. Required the number belonging to logarithm 5.345678. 345570, The first four figures of the natural number are 2216. With the tabular difference 196, the fifth figure, corresponding to 108, is seen to be 5, with a remainder of 10. To find the sixth figure corresponding to this remainder 10, we may multiply it by 10, making 100, and search for 100 in the same line of proportional parts. We see that a difference of 100 would give us 5 in the fifth place of the natural number. Therefore, a difference of 10 must give us 5 in the sixth place of the natural number. Hence the required number is 221655. In the same manner we find the number corresponding to log. 3.538672 is 3456.78; MULTIPLICATION BY LOGARITHMS. (11.) According to Art. 3, the logarithm of the product of two or more factors is equal to the sum of the logarithms of those factors. Hence, for multiplication by logarithms, we have the following RULE. Add the logarithms of the factors; the sum will be the logarithm of their product. Ex. 1. Required the product of 57.98 by 18. The logarithm of the product 1043.64 is 3.018551 Ex. 2. Required the product of 397.65 by 43.78. Ans., 17409.117. Ex. 3. Required the continued product of 54.32, 6.543, and 12.345. The word sum, in the preceding rule, is to be understood in its algebraic sense; therefore, if any of the characteristics of the logarithms are negative, we must take the difference between their sum and that of the positive characteristics, and prefix the sign of the greater. It should be remembered that the decimal part of the logarithm is invariably positive; hence that which is carried from the decimal part to the characteristic must be considered positive. Ex. 4. Multiply 0.00563 by 17. The logarithm of 0.00563 is 3.750508 Ans, 0.022199. Product, 0.09571, whose logarithm is 2.980957. Ans., 0.00003738. Ex. 7. Find the continued product of 11.35, 0.072, and 0.017. (12.) Negative quantities may be multiplied by means of logarithms in the same manner as positive, the proper sign being prefixed to the result according to the rules of Algebra. To distinguish the negative sign of a natural number from the negative characteristic of a logarithm, we append the letter n to the logarithm of a negative factor. Thus the logarithm of -56 we write 1.748188 n. Product, -1575.47, log. 3.197409 n. Ex. 9. Find the continued product of 372.1, -.0054, and -175.6. Ex. 10. Find the continued product of -0.137, -7.689, and -.0376. DIVISION BY LOGARITHMS. (13.) According to Art. 3, the logarithm of the quotient of one number divided by another is equal to the difference of the logarithms of those numbers. Hence, for division by log. arithms, we have the following RULE. From the logarithm of the dividend, subtract the logarithm of the divisor; the difference will be the logarithm of the quotient. Ex. 1. Required the quotient of 888.7 divided by 42.24. The logarithm of 888.7 is 2.948755 The quotient is 21.039, whose log. is 1.323031. Ex. 2. Required the quotient of 3807.6 divided by 13.7. Ans., 277.927. The word difference, in the preceding rule, is to be understood in its algebraic sense; therefore, if the characteristic of one of the logarithms is negative, or the lower one is greater than the upper, we must change the sign of the subtrahend, and proceed as in addition. If unity is carried from the decimal part, this must be considered as positive, and must be united with the characteristic before its sign is changed. Ex. 3. Required the quotient of 56.4 divided by 0.00015. The logarithm of 56.4 is 1.751279 The quotient is 376000, whose log. is 5.575188. This result may be verified in the same way as subtraction in common arithmetic. The remainder, added to the subtrahend, should be equal to the minuend. This precaution should always be observed when there is any doubt with regard to the sign of the result. Ex. 4. Required the quotient of .8692 divided by 42.258. Ans. Ex. 5. Required the quotient of .74274 divided by .00928. Ex. 6. Required the quotient of 24.934 divided by .078541. Negative quantities may be divided by means of logarithms in the same manner as positive, the proper sign being prefixed to the result according to the rules of Algebra. Ex. 7. Required the quotient of -79.54 divided by 0.08321 Ex. 8. Required the quotient of -0.4753 divided by -36.74. INVOLUTION BY LOGARITHMS. (14.) It is proved in Algebra, Art. 340, that the logarithm of any power of a number is equal to the logarithm of that number multiplied by the exponent of the power. Hence, to involve a number by logarithms, we have the following RULE. Multiply the logarithm of the number by the exponent of the power required. Ex. 1. Required the square of 428. The logarithm of 428 is 2.631444 2 Square, 183184, log. 5.262888. Ex. 2. Required the 20th power of 1.06. 20 20th power, 3.2071, log. 0.506120. Ex. 3. Required the 5th power of 2.846. It should be remembered, that what is carried from the decimal part of the logarithm is positive, whether the characteris tic is positive or negative. Ex. 4. Required the cube of .07654. · The logarithm of .07654 is 2.883888 3 Cube, .0004484, log. 4.651664. Ex. 5. Required the fourth power of 0.09874. EVOLUTION BY LOGARITHMS. (15.) It is proved in Algebra, Art. 341, that the logarithm of any root of a number is equal to the logarithm of that num Der divided by the index of the root. Hence, to extract the oot of a number by logarithms, we have the following B |