$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 50; and 50253 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. 0.12 1.079181 = log. 0·0043.602060 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 38; 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. 365 = 2·562293 log. 7.3 0.863323 quotient is 5, its log. = 1.698970 2. What is the quotient of 2.456 by 1·47? Ans. 167075, nearly. 3. What is the quotient of 7.4 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? 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. 1431363, 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. 0.005673-7535835 +2.753583, which divided 3. What is the 3d root of 0.365? Ans. 0.714657, nearly. Ans. 1-32047, nearly. $31. Proportion by Logarithms. Since the fourth term of a proportion is found by dividing the product of the second and third terms by the first, we obviously have this RULE. From the sum of the logarithms of the second and third terms, subtract the logarithm of the first term. What is the fourth term of a proportion of which the first, second, and third terms are respectively 0-0146, 45, and 1.07? subtract log. 0.0146 = 2.164353 log. of fourth term = = 2.518244 Consequently the fourth term = 329.794, nearly. ARITHMETICAL COMPLEMENT. § 32. The subtraction of a logarithm from the sum of two or more logarithms is usually made to depend upon addition, by making use of its Arithmetical Complement, which is the difference between 10 and the given logarithm, and is readily taken from the table, by beginning at the left hand and subtracting each figure from 9, except the last significant figure on the right, which must be taken from 10. Now it is obvious, that if instead of subtracting a given number from another, we first subtract the number from 10, and then add the result, we shall, after rejecting 10, obtain the true difference. Hence to work a proportion by logarithms, we may use this second RULE. Take the arithmetical complement of the logarithm of the first term, and the logarithms of the second and third terms, and from the sum of the thres reject 10 from the characteristic. |