D the number which is found directly opposite the logarithm already taken out, and multiply it by the figures which were regarded as zeros, pointing off in the product as though these figures were all decimals; add the result thus obtained to the logarithm already found, and it will give the logarithm of the given number. In this way we find log. 365365 5.562727. log. 635536 = 5·803140. log. 704307 = 5·847762. NOTE.-Since the foregoing process of finding the logarithm of a number of more than four places of figures is founded on the supposition that the differences of logarithms of different numbers are to each other as the differences of the numbers, which supposition is not strictly true, it follows that this method can be used only to a limited extent. It ought never to be employed for a number consisting of more than six places of figures. § 24. To find the logarithm of a Vulgar Fraction. Since a vulgar fraction is the quotient of the numerator di vided by the denominator, we may obtain its logarithm by subtracting the logarithm of the denominator from the logarithm of the numerator. Thus, the logarithm of 3 is log. 37-log. 531-568202-1.7242761.843926. 5 3 §25. To find the logarithm of a Decimal Fraction. Since we have, by the property of logarithms, log. 4.5 5 0 =log. 4 log. 45-log. 10=log. 45-1; log. 3.65 = log. 18=log. 365-log. 100=log. 365-2; 3 5 log. 3-754 = log. 154=log. 3754-log. 1000=log. 3754—3; 37 it follows that the decimal part of the logarithm of a decimal fraction is the same as though the number was wholly integral, the only difference between the logarithm of a decimal number and of the number considered as integral is in the characteristic. Hence, take out the logarithm as though the number were integral, and fix a characteristic according to rule under § 19. § 26. To find the natural number corresponding to any log arithm. Seek in the table, in the column 0, for the first two figures of the decimal part of the logarithm; the other four figures are to be sought for in the same column, or in any one of the columns 1, 2, 3, &c. If the decimal part of the logarithm is exactly found, then will the first three figures of the corresponding number be found in the column N, and the fourth figure will be found at the top of the page. This number must be made to correspond with the given characteristic of the given logarithm by annexing ciphers, or by pointing off decimals. Thus the logarithm 5.311754 gives 205000, When the decimal part of the logarithm cannot be accurately found in the table, take out the four figures corresponding to the next less logarithm. Then for the additional figures, subtract this less logarithm from the given logarithm, and divide the remainder, with naughts annexed, by the corresponding number taken from column D. For example, let us seek the number whose logarithm is 1.234567. We find the next less number to the decimal 0·234567 to be 0.234517, which corresponds to 1716. We also find the number in column D to be 253. Hence 0.234567 0.234517 0.234770 0.234517 253 50; and 50 ÷ 253 = 0·198, nearly. So that the number answering to the logarithm 1.234567 is 17.16198, nearly. ARITHMETICAL CALCULATIONS BY LOGARITHMS. § 27. Multiplication by Logarithms. Since the logarithm of the product of two or more factors is equal to the sum of their logarithms, we deduce, for multiplication by logarithms, this RULE. Add the logarithms of the factors, and the sum will be the logarithm of the product. 1. What is the product of 3.65 by 56.3? log. of product = 2·312801, which gives 205-495 for the product. 2. What is the product of 7.8 by 35·3? 3. What is the product of 2.13 by 0·57? Ans. 275.34 Ans. 1.2141. NOTE.-When any of the characteristics of the logarithms are negative, we must observe the algebraic rule for their addition. 4. What is the continued product of 53.7, 0·12, and 0.004? log. 5.37 1.729974 log. 0.12 -1.079181 log. 0·004 = 3.602060 product=0.025776, whose log. = 2·411215 5. What is the square of 37; that is, what is the product of 3.7 by 3.7 Ans. 13.69. 6. What is the cube of 3.8; that is, what is the continued product of 3.8, 3·8, and 3·8 ? Ans. 54.872. $28. For Division by Logarithms, we obviously have this RULE. Subtract the logarithm of the divisor from the logarithm of the dividend. EXAMPLES. 1. What is the quotient of 365 by 7.3? log. 3652-562293 log. 7.3 = 0.863323 quotient is 5, its log. = 1.698970 2. What is the quotient of 2.456 by 1.47? Ans. 1·67075, nearly. 3. What is the quotient of 74 by 12.58? NOTE.-When either or both of the characteristics of the logarithms are negative, we must observe the algebraic rule for the subtraction of the one from the other. 4. What is the quotient of 0.378 divided by 0·45? log. 0-3781.577492 log. 0·45 =1·653213 quotient = 0.84, whose log. 1.924279 5. What is the quotient of 0-10071 by 0-00373? log. 0.100711-003072 log. 0.003733-571709 quotient = 27, whose log. = 1.431363, nearly. $29. Involution by Logarithms. Since the exponent denoting any power of any number expresses how many times this number is used as a factor to produce the given power, it follows that the logarithm of any power is equal to the logarithm of the number repeated as many times as there are units in the exponent. Hence we have this RULE. Multiply the logarithm of the number by the exponent deno 2. What is the 7th power of 0·5? log. 0·5 = 1·698970 7 multiply. power = 0.0078125, whose log. = 3.892790 NOTE.-It must be kept in mind that the decimal part of every logarithm is pos itive, so that, as in the last example, whatever is to carry to the product of the characteristic by the exponent is positive. 3. What is the 30th power of 1·07? 4. What is the 11th power of 0.11 ? §30. Evolution by Logarithms. Ans. 7.6123, nearly. Ans. 0·0000000000285313. Since the exponent denoting a root indicates that the number is to be separated into as many equal factors as there are units in the exponents, we obviously have this RULE. Divide the logarithm of the number by the number denoting the root. EXAMPLES. 1. What is the 11th root of 11 ? log. 11 = 1.041393, which divided by 11 gives 0·094672 for the log. of the root. Hence the root = 1.24357, nearly. NOTE.-When the characteristic is negative, and not divisible by the exponent, we must put with it a sufficiently large negative integer to make it divisible, and then connect with the decimal portion of the logarithm an equally large positive number. This will be best illustrated by the following example. 2. What is the 5th root of 0·00567 ? log. 0·00567 = 3·753583 = 5 + 2·753583, which divided by 5 gives 1.550716 for the logarithm of the root. Hence the root = 0.3554, nearly. 3. What is the 3d root of 0.365? 4. What is the 7th root of 7? 5. What is the 5th root of 0.5? Ans. 0.714657, nearly. Ans. 1·32047, nearly. |