LOGARITHMS. 



LOGARITHMS. 



o* that ol /. Let f * be UM function which number ii of iu 

 logarithm : so Uut *f (log *). If then ud + 4 be logarithm* of 

 * u>d y. and if c be the logarithm of x , theii M * : jr : : * : tf, e + 4 

 mut be Uw logarithm of *. An.1 .r. y, x' anil y' are severally fa, 

 f (+), Brandt (<+*). But */-**>, or 



Let + or x be the number which hai for its logarithm ; then o = 0; 

 and calling x the number in question, we hare 



I 



But by the theorem proved in the article BINOMIAL THEOREM, thii 

 can only be true on the supposition that 9 e + * u *uch a function of 

 c M C', where o it independent of r. Consequently, the number 

 wboee logarithm U e mtut be K 0*. This evidently satisfies the con- 

 ditkx, and the theorem quoted ahowi it to be the only function 

 which satisfies the conditions. 



It U meet convenient to assume 1 u the number x, which ha* for 

 it* logarithm. We hare then the following equation, connecting a 

 number with it* logarithm, 



C 1 "*' = x: 



ao that every number haa a logarithm for any value of o we may take, 

 only it muat be remembered that the same value of c must always be 

 msa The logarithms of all numbers for a given value of c form a 

 lytttm : and c u called the bate of that system. 



Given a system of logarithms, we now inquire how to find the 

 logarithms in any other system. Let A and n be the bases of the 

 systems, and a and 4 the logarithms of any number * in the two bases. 

 Then we have 



A* = x, B* = *, or A = B* ; 



whence 



*, or log B (base A)= 



log a: (base A) 



that is, to turn one system of logarithms into another with any new 

 base, divide every logarithm in the system by the logarithm which 

 there belongs to the new base. 



We now proceed to the method of determining logarithms. In the 

 article LIMIT it is shown, by means of the binomial theorem, that of 

 the two series 



a a' a 



<rV 



O 



27S.4 



2T374 



the second U the zth power of the first. A remarkably wmple oac 

 presenU iUelf, which, in fact, leads to Napier's system of logarithms : 

 It is when a = 1. In this case the first series becomes 



+ 2 



! 



2~78 



1 



3.3.4 



which is very convergent, and is 2718281828, very nearly. This 

 remarkable series is generally denoted by f (sometimes by e, Laplace 

 always uses c for it), and we have 



In Napier's system, then (we shall presently show that this is 

 Nmpier'i system), x is the logarithm of 1 +x + ^x i + ____ ; or the 

 logarithm being given, the number can be immediately found. 



Since the last equation is universally true, for x write log a x x. 

 where log a means log a (base ). The first aide then becomes 



. tot .x. ( or (,"), or a; 



=1 + log 0.1 + 



2 



flog .y. * 



=log+ j T 



if c be diminished without limit, we have then 



a* -1 



Limit of J -" = log (base ) ; 



or, far a given (sod very small) value of z.the logarithms of different 

 numbers () are very nearly in the proportion of the values of a* - 1. 

 This is the theorem to which we have before alluded. 



Let a= 1 + 4, then 



(l+D'-l 



*-l 



"T 



- - 



&+* -^ | 



x-l*-2 



if x diminish without limit, the limit of the first side has been shown 

 to be log (1 + A), the base being t, which U always to be understood 

 when the contrary is not expressed. The limit of the second aide u 

 easily found by making x-(>, and we thug have 



V V 



log(i+i)=&- Y + - + , 



which however Uonly convergent when b lies between - 1 and + 1 

 Since this lust i universally true, we find, by substituting i for 6, 



6 1 fr i< 

 log(l-4)=-4- T - T - -....; 



and subtracting the first from the. second, remembering that 



we find that 



1+4 



1+4 



-log(l-4) 



t> 



4 



lg* 



fx-1 1 /r-l\> -, 



=2 + +. -..,. 



which is always convergent, but converges very slowly when x U 

 considerable. If however we make 



x-l 



z+1 



then, remembering that log = log (j+ 1) log s, we have 



which is very convergent when z U even so small as 1, and serves to 

 find the logarithm of any number when that of the next lower number 

 is given. The two following series, which may be easily proved from 

 the preceding, will complete the list of those which are most useful in 

 practice : 



o 1 o 1 o 



iog(+o)=iQg+ 7 - 2 ? + 3 3 -; 



It only remains to show the identity of this system with that of 

 Napier. If ( be the number of seconds elapsed from the beginning of 

 a motion, and if a* be the length described in that time, then the time 

 is the logarithm of the length described The velocity at the end of ( 

 seconds is the differential coefficient of a', or a '.log a, where the 

 logarithm used is that of the preceding algebraical system ; this 

 velocity U therefore log a at starting, or when = 0. Now, in Napier's 

 system thin velocity U unity, or a = : that U, the base of Napier's loga- 

 rithms is the series called . But in the system whose base is 10, 

 log a is 2 302585 1, which is the velocity at starting assumed by Briggs. 



By the foregoing aeries a system of Naperiau logarithms may be 

 calculated with a very small fraction of the labour which they cost 

 Hi. u inventor. This having been done for all whole numbers within 

 the given limit*, the logarithm of any fraction ia readily found by 

 subtracting the logarithm of the denominator from that of the 

 numerator. 



It rnuBt be admitted that Briggs, by his constmction of the decimal 

 system, divides with Napier the merit of invunting logarithmx, con- 

 sidered as an instrument of calculation. In the Naperian system the 

 table must either be carried to an enormous length, or whole numbers 

 only must have logarithms, and every logarithm of a lr.i--ti..n will 

 require two entries of the table and a subtraction. But in Brigga's 

 M -MII the logarithm of every decimal fraction can be found by one 

 entry of the table, and one inspection of the fraction. 



The peculiarity of this system (the base of which is 10) is as follows : 

 number or fraction is either a power of ten, positive or negative, 

 or lies between two powers of ten. The powers of ten are ranged in 

 the following table : 



10- = -0001 10 = 1 



10-> = -001 

 10- -01 

 10-' = -1 



10' - 10 

 10' - 100 

 10 = 1000 

 10* = 10000 



