0, 1, 2, 3, 4, &c. Thus, for all numbers lying between 100 and 1000, the corresponding power of 10 being greater than 2, and less than 3, must be 2, increased by some decimal fraction; for numbers lying between 1000 and 10000, must be 3, increased by some decimal fraction; and so on. The logarithms, therefore, of intermediate numbers consist of an integer and of a decimal part; the decimal part alone is registered in the Tables, and is called the Mantissa; the integer part, which is called the Characteristic, is not entered, but may be found by RULE I. The characteristic of the logarithm of a number greater than unity is one less than the number of the digits of its integer part. Thus, the characteristic of the logarithm of 849 is 2; for as this number lies between 100 and 1000, its logarithm must lie between 2 and 3. The mantissa, by referring to the Tables, is found to be 92891; therefore log 849 = 2.92891 The logarithms of numbers less than unity are negative, as may be seen from the following table : 10-1 Ib = ido Tobo 10 Todoo &c. &c. It follows from this, that the characteristics of the logarithms of all numbers less than unity are negative, and may be found by = 0.1 RULE II. The characteristic of the logarithm of a number less than unity, and reduced to the decimal form, is negative, and one greater than the number of ciphers following the decimal point. A negative characteristic is denoted by writing over it the negative sign; thus, ī, 2, 3, &c. The reason of the rule may be seen from inspecting the preceding table of values of negative powers of 10, or perhaps more clearly by considering a particular case; for example, log 849 = 2.92891 therefore 849 = 102.92891 If we divide each side of this equation successively by 10, we obtain, 84.9 = 101.92891 8.49 = 100.92891 from which it follows that Jog 84.9 = 1.92891 log 8.49 = 0.92891 This corresponds with the rule given for determining the characteristic of the logarithm of a number greater than unity. If we continue the division of each side by 10, we obtain 0.849 107.92891 102.92891 &c. &c. From which it follows that log 0.849 = 1.92891 &c. From this it appears that the negative characteristics are given by the rule in question. It also appears that the logarithms of all numbers, consisting of the same significant figures, have the same mantissa ; and that the characteristic depends solely on the position of the decimal point. 3. Properties of Logarithms.-We shall now demonstrate four general propositions, from which the rules for using logarithmic tables in numerical computations are derived. PROPOSITION I. The logarithm of the product of two numbers is equal to the sum of the logarithms of the numbers. If the numbers be N and M, let n = log N, and m = log M to any base a, then by the definition, N an sam By multiplication, NX M = antm therefore, log N M = n + m = log N + log M. PROPOSITION II. The logarithm of the quotient of two numbers is equal to the difference of the logarithms of the numbers. By division. N M therefore N = n - m = log N – log M M = arm log PROPOSITION III. The logarithm of the pth power of a number is equal to p times the logarithm of the number. If we raise to the pth power each side of the equation The logarithm of the pth root of a number is equal to the pth part of the logarithm of the number. If we take the pth root of each side of the equation N = an we obtain NBC therefore log N log NP = р n 4. Logarithmic Tables.-If the number be given, its logarithm may be found by the following rules. If the number consist of less than five figures, we proceed by RULE III. With the given number enter the column marked N; and opposite will be found the mantissa. To this prefix the characteristic. The result will be the required logarithm. EXAMPLES. 2. 1. Find the logarithm of 3562. 5621. 3. 832.5. 4 13:54. Ans. 3.55169. Ans. 3.74981. Ans. 2.92038. Ans. 1.13162. If the number consist of more than four figures, RULE IV. Find the mantissa corresponding to the first four figures. With the additional figures, one by one, enter the column PP under the proper tabular difference, and add the corresponding proportional parts to the mantissa first found. Prefix the characteristic. The result will be the required logarithm. log 189637 = 5.2779284 84 log 20.3643 = 1.3088778 If the logarithm be given, the number which corresponds to it is found by the following rules. When the mantissa can be found in the tables, we proceed by RULE . 1°. With the given mantissa enter the column marked Log. 2°. Take from the column N the corresponding four figures. 3o. From the characteristic, find the position of the decimal point. The result will be the required number. EXAMPLES. 1. Given the logarithm 2.93120; find the number. Ans. 853.5. 2. Given the logarithm 1.72534; find the number. Ans. 53.13: 3. Given the logarithm 5.54986; find the number. Ans. 354700. 4. Given the logarithm 2.85986; find the number. Ans. 0.07242. 5. Given the logarithm 0.54741; find the number. Ans. 3.527 When the mantissa cannot be found in the tables, we proceed by RULE VI 1°. Find the next lower mantissa, and note the four corresponding figures in column N. 2°. With the difference between this and the proposed mantissa, enter the table of proportional parts, and find the additional figures. 3o. Place the decimal point according to the characteristic. The result will be the required number. |