LOGARITHMS.] 



MATHEMATICS. ALGEBRA. 



517 



And o'+ = 1 +A(x+y)+B (x+yY+C(x+y) 3 + ... 

 = 1 + Ax + Bx 2 + Cx 3 + Dx< + 

 + Ay + 2B.ri/ 



+ By 2 + SO*/ + 6Vx 



(/) 



Now, from the early part of this article, it appears 

 that there is one definite expansion of a* , and . -. of a' + > 

 Hence (e) and (/), which are each the expansion of a* + , 

 are not merely equal, but are actually identical ; there- 

 fore they must be, term by term, the same. 



AB = 30 



Hence, 

 a* = l 

 where 



C 



A.V . Ax* 







1-2 -3 -4 



(VI.). 



N.B. Suppose e to be such a number that 



i _(,_i)_ <=!)? + .<L=!) &=!>_+.... 



Thene* = l + x- r - T 2 + r 2^ + r 2^4 + ---- 



This number e is very important, and is called the 

 base of the Napierian Logarithms, for reasons to be ex- 

 plained hereafter. We can easily ascertain its value in 

 the following manner : In the above series let x = 1. 

 Then 



1.1.1.1. 



r 1-2 



-2 1-2 3 1-2-3 -4 

 This series is convergent, since it is clearly 



The series, therefore, gives the numerical value of e. 

 This value may be calculated as follows : 



2 l-OOOOOOOOOQ 



10 



n 



500000000 

 16C66660C 

 0416GGG66 

 008333333 

 001388888 

 000198412 

 000024801 



000000275 



12 O00000025 



13 000000002 

 | -000000000 



These decimals are respectively -j-^ j.g.jj 1 .. . 1-2 -3 -4' 



* , &c. Their stun is 718281827 ; and hence e => 



2 7182818, which is quite accurate, so far as it goes. 



The student will observe, that the reasoning in the 

 above article is founded upon the assumptions (1), that 

 a* can be expanded in a series of ascending powers of x ; 

 (2), that it can be expanded in only one series of that 

 kind. These assumptions, in the present case, may be 

 considered as resting on the fact that the expansion of 

 a* is simply a transformation of the binomial theorem. 

 The same remark applies to the following article : If 

 we make the assumption general viz. , that every func- 

 tion of x can be expanded in a single series of ascending 

 powers of x, we enter upon a question which has given 

 rise to many discussions, which cannot be further noticed 

 here. 



Definition. If a = x, then y is called the logarithm 

 of x to the base a, and is generally written y = log. n x 

 where logo means "logarithm to base a." 



12. To show that A log.. (1 + x) =]x ~ + ~ * 



+ etc., where A has the same value as in the last article. 



This is called the logarithmic series. 



Let y log. a (1 + x). 



Then by definition 1 + x = of- 



.-.(! + *)-" 



3.2 3.1 

 By the last article, if fc = x ~2~~f~~3 -- &- 



1-n- 



and a" = 



. 

 A.yn + -.%- 



Now, since (1 + x)" is identically the same as a"", these 

 two series must be identically the same. 



+ - H rvrn 



' ^l 1 I I * *-*) 



where A = (o 1) * ^ ~f o &- 

 N.B. Now 1 = (e 1) fc-j-2 + (e ~ 1)5 &o. 

 .-. log.(l + x)=x y + y y+ <tc. 



/_ i\ /_ 1x3 



<tc. 



-I" -3-^ 

 .. A= log./j. 

 Hence, equation (VII.) may be written 



log./ log.. (1 + x) = x ** + ^ ^ + 

 It will be observed that if x be < 1, the series 



~3 ~3 



.fee. 



is term by term less than x x* + x 3 (fee. 

 and is therefore convergent, provided x is < 1. 



ON THE CALCULATION OF LOGARITHMS. 



13. Oft the Calculation of the Arithmetical Values of 

 Quantities expressed by Infinite Series. 



In the Treatise on Elementary Algebra, the method 

 has been explained of obtaining in numbers the value of 

 an algebraical expression, when definite values are as- 

 signed to the letters composing the expression.* For 

 instance, if o = 2 and 6 = 5, then (2a -f- 6) (6 a)= 27. 

 The student may ask, how can an infinite series be re- 

 duced ? Although we have already given three instances 

 of the manner of doing this, the question is well worth a 

 distinct consideration. We have already seen that 



Now, if x = |, the fraction is equal to 2 ; and we know 

 that if we took the whole number of terms of the series 

 we should get exactly 2. The first two terms are 1-6 ; 

 the first throe 1-75 ; the first four T876 ; the first five 

 terms 1-9375; the first six 1 '96875; each result being 

 nearer to the truth than the one before. Thus, by taking 

 a sufficiently large number of terms, we can get as near 

 to the exact value as we like. The series, in fact, affords 

 the means of approximating to the true value. Of course, 

 in such a case as the above, we should not care for the 

 approximation, since we can so readily get the real value. 

 But in the large majority of cases we cannot get at the real 

 value, or even the real value cannot be expressed by digits 

 at all i.e., is not commensurable with unity; in such cases 

 See ante, p. 491. 



