LOOAHITH11S, USE OF. 



LOGARITHMS, USE OF. 



S3" 



third operation* in the following aaceoding Males by means of the 

 > ea*y oo* which precede* it, aided by consultation of tables :- 



FfHfir of Po<rtn. 



! ... 



SvMnwuaa, 



Dirteloa, 



Tbiu multiplication u reduced to addition. raiting of powers to unilti 

 pBfg^n diruiua tu subtraction, extinction of ruote to division. 



i The mm of the lo,-nUiui of two uuiubciii give* tlie logarithm 

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

 the quotient, u>d o on. Theea ralei are beat expressed in symbols, 

 thus: 



lug A + log B . log (A x B) 



log A - log log (A-TB or '- J 

 log (A") = xlogi 



Thus the characUrUtic of 2791-68 is 3 ; that of 17462 U 4 ; that o! 

 20 137 is 1 ; that of 9-009 is ; that of 763 id T; that of '0198 is 2 : 

 that of 000072 is 8". 



8. It is worth while to remark that this broken rule, u it seems to 

 be, requires subdivision only on account of the notions generally 

 attached to the drrimal / '. .\ lii.-li is treated as if it were one of th> 

 places of the number. But if tlie ilim.il -vint were, an it ou 



be, considered as part and parcel of the unit'i place, so tli.it 1 - 

 instance, is not | 1 | 2 | | 8 | 4 | , but | 1 , 2- | 3 I I .thru the two 

 nileemightbegiven under one, as follows. Thecharan number 



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

 from the unit's place ; ami \ positive when that first to tin 



It it. negative when tu the right TIUIB in 1234 567 or 12o (4-) 56711 is 

 3 ; in -00029 or (0-) 00029 it . 



9. A mixed niirnW. -m-li < : ' '.'-T41, is thuH multi]>li 

 divided : 



3- H2741 J7I1 



3. Tlie logarithm! wanted are taken pertly from * table, partly from 

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

 found by the rule ; the fractional part by the table. 



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

 the number : thu 2167-3 has 3 33592 for its logarithm, and 3 u the 

 characteristic of 3167*3. It it very common tu call this 3 the charac- 

 teristic of the logarithm itaelf ; 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 rent, or fractional part, of 

 Uw logarithm of that number depends upon the significant figures 

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

 those figures gain by their position relatively to the decimal point. 

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

 with different characteristics, or different integer portions in their 

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

 of all. namely, -33592 



0. The characteristic of a number may be either rxxi'ire or xa/atiir. 

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

 using these quantities, directing the reader who is 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. Tlie following examples will illustrate 

 this ; the negative quantity being distinguished from the positive one 

 by a bar drawn above it : 



B 



;-,0446 



1- 65457 



! In multiplication, tin- multiplication of the negative figure produces a 

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



nution of this mini bur. The lust step is : 6 times 3 is lo, and 5 is 13. 



In division, a divisible figure must be sought attire the negative 

 ; characteristic, not bttoa it ; and the units 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 5, the first step is : 5 ; 



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



8 and 5 make 3. 

 8 and 11 ., 

 2 and 7 .. K 

 4 times ti in - 1 ; 

 if we carry 3, it gives 21. 



2 from 3 gives 1. 



2 from 3 .....". 



3 from '2 



U from 5 . . . . 1 . 

 8 from 3 . . , . ft 



6JIHW 



i -'.'-j 



lii)3I-606 

 4-961 



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

 are significant figures before the decimal point, the characteristic is 

 one less than the number of them. But when the significant figures 

 begin after the decimal point, the characteristic must be marked 

 and must point out the jJact in tckick eiyn(t<caiicc tryins. 



In. The decimal part of the logarithm is taken out of the tubl<je. 

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

 the real logarithms are generally interminable decimals. The only 

 numbers which have logarithms capable of finite expression (hi the 

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

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

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



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

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

 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 that, so 

 many decimals as there are in the logarithm*. 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 

 001286, -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 tlrnn under stated in the above. Four figures are very often 

 enough ; five figures almost always. When five figures arc not sufficient, 

 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. 



FOUR-FIGURE T.vBLt. 



KivE-noi'RE TABLE. 



