LOGARITHMS. USE OF. 



LOGARITHMS, USE OF. 



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third operation* in the following ascending scale* by mean* of the 

 met* ay MM which precede* it, aided by consultation of table* : 



Multiplication, 

 Division, 



tabtncUon. 



TMn|c of r-o 

 Eilnction of RooU. 



Thu* multiplication U reduced to addition, raiting of powers to uiulti- 

 rjhnn division to aubtractiun, extraction of rooU to division. 



t The mm of the logarithms of two number* gives the logarithm 

 of their product, the difference of two logarithms give* the logarithm of 

 the quotient, and to on. These rule* are belt expressed in symbol*, 

 thus: 



l< -g A + l>jg B = log (A x B) 



log A - log B = log (A-T-B or '- J 

 log (A ) = *t x log A 

 log ( /*) = Io 8 A -5-*- 



3. The logarithms wanted are taken partly from a table, partly from 

 an easily remembered rule. The integer part of the logarithm is 

 found by the rule ; the fractional |rt by the table. 



4. The integer portion of the logarithm is called the eharacterlilie of 

 the number : thus 2167'3 has 3-33592 for its logarithm, and 3 is the 

 characteristic of 2167*3. It is very common to call this 3 the charac- 

 teristic of the logarithm itself ; but it is better to fix this appellation 

 upon it in connection with the number from which it arises. 



5. The characteristic of a number depends entirely upon the place 

 which the decimal point occupies. But the rest, or fractional part, of 

 the logarithm of that number depends upon the significant figures 

 only of the number, and has no connection with the meaning which 

 those figure* gain by their position relatively to the decimal ]>oint. 

 Thus -21673, 21-673, -00021673, 21673. 21673000, Ac., are numbers 

 with different characteristics, or different integer portions in their 

 logarithms. But the fractional portions are the same in the logarithms 

 of all, namely, -33592 



6. The characteristic of a number may be either jxaitite or n 



For our present purpose, it will be sufficient to lay down the rules for 

 tuing these quantities, directing the reader who U not used to the 

 distinction to try to explain them by considering positive quantities 

 as gains, negative quantities as losses, addition as junction, subtraction 

 as removal. Thus the addition of a positive quantity (annexation of a 

 gain) is the same thing as the subtraction of a negative quantity 

 (removal of a loss) ; and so on. The following examples will illustrate 

 this ; the negative quantity being distinguished from the positive oue 

 by a bar drawn above it : 



Thu* the characteristic of 2791-63 is 3 ; that of 17462 U 4 ; that of 

 29 187 U 1 ; that of 9-999 is ; that of 73 is I; that of '011'- 

 that of 000072 U 5". 



8. It is worth while to remark that this broken rule, aa it K< 



be, require* subdivision only on account of the notions general!} 



attached to the dcrimal / '. xvhieh is treated as if it were one of the 

 j places of the number. But if the decimal point were, .is it < 

 ; be, considered as part and parcel of the uuit'i jJace, so that 1. 



instance, is not | 1 | 2 | | 3 | 4 | , but | 1 | 2- | 3 I 4 . then the two 

 : rules might be given under one, as follows. The characteristic of a mi'i.l>er 



is the number of places by which the first significant figure is distant 



from the unit's place ; and ix positive when that first 



left, negative when to the right. Tims ill 1234 5ti7 or 12o (4') 567 it U 



3 ; in -00029 or (0') 00029 it i 



9. A mixed mimW. MI. !> .,"11, is thus multiplied 

 divided : 



~5- 27-ll 

 ; 



1- 65457 



13- T.0446 



In multiplication, the multiplication of the negative figure produces a 

 negative result, and the carriage from the positive part goes in dimi- 

 nution of this number. The last stop is : 6 times 8~is lo, and 5 is 13. 

 In division, a divisible figure must be sought attire tin- negative 

 characteristic, not bdmr it ; and the unite necessary to make up that 

 divisible figure must be afterwards carried to the right in the usual 

 way. Thus when 21-116 is divided by B, the first step is : 5 is con- 

 tained in 25, 5 times, carry 4 ; then 5 is contained in 41, 8 times, &o. 



6J5H16 



1-292 



111)31-606 

 4-961 



8 and 5 make 3. 

 8 and 11 ., 

 2 and 7 



4 times 6 is - 1 ; 

 if we carry 3, it gives 21. 



2 from 3 ? 



2 from 3 .... .1. 



3 from 'I . . . . ... 



U from 3 .... 1 . 



8 from 3.... .'. 



7. The rule for finding the characteristic is as follows. When there 

 are significant figures before the decimal point, the characteristic is 

 one lea* than the number of them. But when the significant figures 

 begin after the decimal point, the characteristic must be marked 

 negative, and must point out tin ji'ucc in >'lii-h liynlji&mcc bnjins. 



In. The decimal part of the logarithm is taken out of the 

 These are not complete logarithms, which could not be given, tiuce 

 the real logarithms are generally interminable decimals. The only 

 numbers which have logarithms capable of finite expression (tn the 

 system commonly used) arc those in the series .... '0001, -001, "01, 1, 

 1, 10, 100, 1000, .... of which the complete logarithms are the charac- 

 teristics themselves, or 4, 8, 2, 1, 0, 1, 2, 3, The tables give 



only a certain number of the first decimal places, and may be < 

 after the number of decimals in the logarithms they give. Tims a 

 table which gives four decimals in each logarithm may be called a 

 four-figure table ; and so on. A calculator chooses his table according 

 to the degree of accuracy he wants: the general rule being lli 

 many decimals aa there are in the logarithms, so many significant 

 figures of the answer may be found correct, with perhaps an error of a 

 unit or two in the last figure. Thus if the real answer to a question 

 were -00123769728, we might expect from four-figure tables to get 

 001236, -001237, '001238. or -001239. But seven figure tables would 

 probable give from '001237695 to -001237699. Perhaps the liability 

 | of the last figure to error is, for the general run of questions, rather 

 over tlmn imiler stated in the above. Four figures ore very often 

 enough; five figures almost always. When five figures arc not sutl 

 we should recommend having recourse to seven at once, for a reason 

 presently mentioned. 



11. We insert specimens of a four-figure, five-figure, and a seven- 

 figure table. 



FUUB-FIGURE TABLE. 



FIVK-FIOTIIE TABLE. 



