$23. To find the logarithm of a number consisting of five or more places of figures. For example, let us seek the logarithm of 734582. We readily find as follows: = Diff. of logs.=59: The difference between the first number 734500 and the given number is 734582-734500 82. Hence, if we suppose the difference of logarithms to be to each other as the difference of their corresponding numbers, which supposition is very nearly correct, we shall have 100 82 59: 48-38, for the difference between the logarithm of 734500 and the logarithm of 734582. Hence, we have Had there been only five figures in the given number, that is, only one beyond the fourth place, the first term of the above proportion would have been 10 instead of 100; had there been three additional figures beyond the fourth place, the first term would have been 1000. The difference between the logarithms of consecutive numbers of four places of figures is given in the table in a column headed D. Thus by turning to the table, we find 59, our third term of the above proportion, immediately opposite the loga rithm of 7345. From what has been done, we see that we may find the logarithm of a number consisting of more than four places of figures by the following method: Consider all the figures after the fourth as zeros. Then find the decimal part of the logarithm of the number given by the first four figures, observing to give a characteristic for the whole number of figures by rule under § 19. Take from the column 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. 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. 53=1.568202-1.724276 = 1.843926. $25. To find the logarithm of a Decimal Fraction. Since we have, by the property of logarithms, log. 3.754 = log. 115=log. 3754-log. 1000 = log. 3754-3; 375 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 logarithm. 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 1234567. 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 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. 56.31.750508 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 3·7; 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. 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 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. 0.10071 I-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 denoting the power. EXAMPLES. 1. What is the 5th power of 1.234 ? log. 1.234 0.091315 = power = 2.86137, whose log. = 0·456575 5 multiply. |