This number is usually denoted by the letter e; and logarithms to this base are called Napierian logarithms, from Napier the inventor of logarithms. This system of logarithms, although very important in theory, is not used in practical calculations; and we shall not require to consider it in the present work. In the other system the base is 10; this system is used in practical calculations, and is called the common system. 58. We shall not in the present work explain how a table of logarithms is calculated; for this the student may refer to the larger treatise. We may remark that in very few cases can a logarithm be assigned exactly, but as close an approximate value as we please can be found; for example, a table may be constructed which shall give logarithms to seven places of decimals. We shall shew in the next three Articles what are the chief advantages of the common system of logarithms. 59. In the common system of logarithms if the logarithm of any number be known, we can immediately determine the logarithm of the product or quotient of that number by any power of 10. For log10 N× 10′′=log10 N+log10 10′′ = log 10 N+n ; log N 10%10 10 =log10 N-log10 10" = log10 N-n. That is, if we know the logarithm of any number we can determine the logarithm of any number which has the same figures, but differs merely by the position of the decimal point. In future we shall for brevity use log for log10, that is we shall omit to specify the base 10. 60. We know from Arithmetic that 10o= 1, 101=10, 102=100, 103=1000,... Now from this we infer that if a number lies between 1 and 10, its logarithm lies between 0 and 1; if a number lies between 10 and 100, its logarithm lies between 1 and 2; if a number lies between 100 and 1000 its logarithm lies between 2 and 3: and so on. The integral part of a logarithm is called the characteristic, and the decimal part is called the mantissa; thus as the logarithm of any number between 100 and 1000 is greater than 2 and less than 3, it is equal to 2+ some decimal thus in this case 2 is the characteristic. : We shall now give an important proposition respecting the characteristic. 61. In the common system of logarithms the characteristic of the logarithm of any number can be determined by inspection. For suppose the number to be greater than unity, and to lie between 10" and 10"+1; then the logarithm is greater than n and less than n + 1, so that the characteristic of the logarithm is n. Next suppose the number to be less than unity, and to lie between and 1 10" 1 10"+19 that is between 10- and 10--1; then the logarithm will be some negative quantity between -n and (n+1); hence if we agree that the mantissa shall always be positive, the characteristic of the logarithm will be -(n+1). Hence we have the following rule: the characteristic of the logarithm of a number is one less than the number of integral figures of the number; when the number has no integral figures the characteristic of the logarithm is negative and is one more than the number of cyphers immediately to the right of the decimal place in the number. 62. By reason of the properties explained in the three preceding Articles it is unnecessary in a table of common logarithms to print either the characteristics of the logarithms or the decimal points of the numbers. For example, we find in a table the following figures: Number. Logarithm. This means that 1938756 is the mantissa; for the number 15627 the corresponding characteristic is 4, and therefore log 15627=4.1938756. Similarly log 156-27=2.1938756, and log 0015627-3-1938756: in the last example 3 is equivalent to -3, so that we express in the manner indicated the fact that log 0015627-3+*1938756. 63. It is necessary to notice one point in practical operations with negative characteristics. Suppose we require the logarithm of the cube root of 0015627. By Art. 55 the logarithm is 3 of 3·1938756. The division here can be immediately effected; for of -3 is -1; and of 1938756 is '0646252; thus the required logarithm is 1.0646252. But suppose we require the logarithm of the square root of ⚫0015627. By Art. 55 the logarithm is of 3.1938756. It is convenient now to put 31938756 in the form −4+1·1938756; then dividing by 2 we obtain−2+*5969378, so that the required logarithm is 2.5969378. Similarly if we require the logarithm of the sixth root of 0015627 we put 3·1938756 in the form -6+3 1938756; then dividing by 6 we obtain −1+5323126, so that the required logarithm is 15323126. 64. The following example will illustrate the present Chapter. Given log 3='4771213 find the log of (2.7)3 x (81) (90) Let N denote the given expression; then 5 4 3 log 3-1, log =4 log 3-2, 102 log 90=log 32 × 10=2 log 3+1; hence log N= 4 3 (3 log 3 − 1) + — (4 log 3 − 2) — — (2 log 3+1) 5 5 4 8 9+ 5 5 4 117_2.7780766 nearly. EXAMPLES. V. Find the logarithms of the following six numbers to the assigned bases: Given log 2=3010300, log 3=4771213, find the logarithms of the following twelve numbers: 15. 375. 16. 03. 17. 6-3. Given log 3=4771213, log 7=8450980, find the logarithms of the following three numbers: 18. (5)-1. Given log 8=9030900, log 9=*9542425, find the logarithms of the following three numbers: 22. 13. 23. 1. 24. 1. 25. Given log 8='9030900, log 27=1'4313638, find log 21. 26. Write down the characteristics of the logarithms of 34512, 345·12, 034512, and 000034512: also having given log 34512=1.5379701, find the logarithm of the product of the above four numbers. 27. The decimal part of the log of 36541 is 5627804, find the log of 15/('000036541). 28. Find the log of '0625 to the base 8. 29. Given log 1'4=1461280, log 1.5=1760913, find log 000315. 30. Given log 2 find the log of 50 to the base 25. VI. Use of Tables. 65. Many collections of Mathematical Tables have been published, differing in extent and in the number of decimal places to which they are carried; and thus practical calculators are enabled to provide themselves with such Tables as are most convenient for the special work on which they may be engaged. A collection of Tables published by W. and R. Chambers may suffice for ordinary purposes. A cheap and very extensive collection of Tables has been edited in Germany by Schrön, and this work has been introduced into England with a Preface by Professor De Morgan. 66. Collections of Tables usually contain explanations of the mode in which they are arranged, together with instructions for using them. We shall accordingly only give here some examples which will suffice to guide the student who may wish to use any Tables for occasional calculation. We shall not give investigations of the accuracy of the methods which we exemplify; for such investigations the student is referred to the larger treatise. 67. One general consideration which applies to the use of Mathematical Tables is this: we rarely find what we require immediately in the Tables, but we find two entries between which what we require must lie, and from which it must be determined. Accordingly we have to exemplify the method of proceeding in such cases. 68. To find the logarithm of a given number. If the given number is contained in the Table we take the decimal part of the logarithm from the Table, and prefix the characteristic; see Art. 61. Suppose however that the number is not contained exactly in the Table. The Table, for example, may give the logarithms of all numbers from 1 to 100000, and we may require the logarithm of 5632147. |