tTSE OF LOGARITHMS.] 



MATHEMATI CS. ALGEBRA. 



619 



And. in point of fact, the inventor of logarithms, Napier, 

 actually calculated logarithms to this base e, which is 

 hence called the base of the Napierian logarithms. 



For ihe purposes of numerical calculation, however, 

 the base 10 possesses the following decisive advantage 

 over any other. 



Suppose 10* = N 



Then 10' + -= NX 10-- 



Now, suppose n to be a whole number, then N. X 10" 

 has the same digits as N in the same order, and only 

 differs from it in having its decimal point shifted n places 

 to the right ; and again 



And when n is a whole number, _ only differs from N 



in having its decimal point shifted n places to the left. 

 It follows, therefore, that the decimal part of the loga- 

 rithm of a number is the same wherever the decimal 

 point may be in the number, and that for every place 

 that the decimal point in the number is shifted to the 

 right, 1. is added to the logarithm ; and for every 

 place, it is shifted to the left, 1 . is subtracted from the 

 logarithm. Thus the table gives us 



So again 



log. 7-5684 

 log. 75-684 

 log. 756-84 



log. -75684 



8790041. 

 1-8790041. 

 2-8790041. 



1 + -8790041. 



or, ax it is more generally written 



log. -75684 - 1-8790041. 



Similarly 



log. -00075684 = "4-8790041. 



It is plain, then, that one calculation gives us the 

 logarithm of the above five numbers, and in fact of as 

 many numbers as can be made by shifting the decimal 

 point to different positions in the combination 75684 ; 

 but if we adopted any other base, we should require a 

 separate calculation for each of them. This advantage, 

 which the base 10 has over any other, was first seen 

 and applied by Briggs, who was professor at Oxford 

 about the year 1670 ; the logarithms are, therefore, some- 

 times called the " Briggian Logarithms." 



The student will perceive that the base 10 has this 

 advantage, in consequence of our system of notation 

 being decimal. If our system were duodecimal, our 

 logarithms would then have to be calculated to the base 

 12, to be possessed of a like advantage, and so on for 

 any other system. 



N.B. The decimal part of a logarithm is called the 

 mantitM : the whole number is called the characteristic. 



18. To thoio that every number hat a Calculable Logarithm. 

 It will be observed that 



may not be (for aught we have yet shown) convergent, 

 unless a < 2. Now, if a be any number, a root of it can 

 always be found, say the no., which shall be less than 2. 

 Now, 



log.- 4- -_ 



which, since a H 1 is less than 1, is convergent. Now, 

 log.' a. n log., o,. . log.' a= n {(a > 1) ^ ^ a " 1) 3 > 



+ ,.("" I) 3 "fe c -)} which series being convergent, if 



n be properly chosen, for every value of a, the value of 

 log., a might be found from it, for all values of a. To 

 calculate a table from such a formula would be most 



laborious ; but as the calculation of the common tables 

 presupposes that we know log., 10, if we treat the sub- 

 ject in its logical order, it would be necessary to calculate 

 log.' 10 from this series, before going further. We shall 

 nnd, if we do so, that 



log., 10 - 2-3025851. 

 Hence 



= -434294481. 



In future pages we shall denote this number by /i. It 

 is called the modulus of the tabular logarithm. 



The logarithmic series to the base 10 we have already 

 seen to be 



x* X s 

 log.. 10 log. (1 + x) = x 2 + 3 , <tc. 



x> x 3 ) 



x TJ + 3 <fcc.(- 



.'. log. 



19. To derive from the Logarithmic series, others from 

 which the numerical values of Logarithms may be 

 calculated. 



We have seen that 



log. (1 + x) - n{x -|' + | 3 _ &c. j 



( X* X 1 ) 



:-. log. (1 x) n I x -3 -3 <fcc.> 

 Now log. (1 +*) log. (l z)=. log. r '-%. 



1 ~ * 



log- = 2, 



l+x q 

 Now suppose j - - . . 



qp 



Suppose 7 = p -f- 1. 

 .-. log. - 2 M 



.-. log. 

 + .... 



= log. p. 



VIII. 



From this we can calculate successively the values of 

 log. 2, log. 3, log. 4 as far as we please. 

 Thus, remembering that log. 1 = 0, we have 



log. 





g- 35+ ....j 



log. 3 = log. 2 + 2,, j-J + |- 1 3 + i.-J s + ... [ 



6-75-- } 



log. 4=log. 



[7+3- 73 



In the ordinary tables the logarithms are given calcu- 

 lated to 7 places of decimals. Hence, in making the 

 calculation from the above series we must take in every 

 term >. 00000001 ; or, since each term is greater than 

 the one that comes after it, we must reduce each term to 

 decimals until we find one < -00000001, which we can 

 omit, together with all that come after it ; all that go 

 before it being reduced to decimals and added together, 

 give the required logarithm. Thus, to find log. 11 : 

 here p = 10 . '. 2p + 1 = 21. 



log- 11 - log- 10 + 2^2! + 5 ' 2F +l-5i> + 



