From the logarithm of 3, Take the logarithm of 16, 0.477121, 1.204120. Leaves the logarithm of, or .1875, 1.273001. Ex. 2. The logarithm of is 2.861697. Ex. 3. The logarithm of 2 is 1.147401. To find the Natural Number corresponding to any Logarithm. 29. 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 66 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 328 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 328, 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 the logarithm 5.345678. The next less logarithm in the table is Their difference is 345570. 108. 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 Propor tional 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 the log. 3.538672 is 3456.78; MULTIPLICATION BÝ LOGARITHMS. 30. According to Art. 23, 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 57.98 is 1.763278 is 1.255273 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, 6543, 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 con sidered positive. Ex. 4. Multiply 0.00563 by 17. The logarithm of 0.00563 is 3.750508 Product, 0.09571, whose logarithm is 2.980957. Ex. 5. Multiply 0.3854 by 0.0576. Product 0.022199, whose logarithm is 2.346334. Ex. 6. Multiply 0.007853 by 0.00476. · Ans. 0.00003738. Ex. 7. Find the continued product of 11.35, 0.072, and 0.017. 31. The logarithm of a negative number is an imaginary quan tity. If, therefore, it is required to multiply negative numbers by means of logarithms, we must multiply the equal positive numbers, and give to the product the sign required by the rule of signs in Multiplication. 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 for-56 we write the logarithm 1.748188 n. Ex. 8. Multiply 53.46 by -29.47. The logarithm of 53.46 is 1.728029 For 29.47 we write the logarithm 1.469380 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. 32. According to Art. 23, 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 logarithms, 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 66 66 42.24 is 1.625724 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 quotient is 376000, whose logarithm 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. The logarithm of 66 66 0.74274 is 1.870837 0.00928 is 3.967548 The quotient is 80.037, whose logarithm is 1.903289. Ex. 6. Required the quotient of 24.934 divided by .078541. If the divisor or dividend, or both, be negative, we perform the division by logarithms by using the equal positive numbers, and prefixing to the quotient the sign required by the rule of signs in 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. 33. It is proved in Algebra, Art. 398, 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. The logarithm of 1.06 is 0.025306⚫ 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 deci mal part of the logarithm is positive, whether the characteristic 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. 34. It is proved in Algebra, Art. 399, that the logarithm of any root of a number is equal to the logarithm of that number divided by the index of the root. Hence, to extract the root of a number by logarithms, we have the following RULE. Divide the logarithm of the number by the index of the root required. Ex. 1. Required the cube root of 482.38. The logarithm of 482.38 is 2.683389. Dividing by 3, we have 0.894463, which corresponds to 7.842, which is therefore the root required. Ex. 2. Required the 100th root of 365. Ans. 1.0608. When the characteristic of the logarithm is negative, and is not divisible by the given divisor, we may increase the characteristic by any number which will make it exactly divisible, provided we prefix an equal positive number to the decimal part of the logarithm. Ex. 3. Required the seventh root of 0.005846. The logarithm of 0.005846 is 3.766859, which may be written 7 +4.766859. Dividing by 7, we have 1.680980, which is the logarithm of .4797, which is, therefore, the root required. |