620 



MATHEMATICS. ALGEBRA. 



[CALCULATION OF LOGARITHMS. 



Now, 

 i - 



J. ^, - -000035993. 

 - 000000040. 



047655089 

 and 2u - -868688962 



..2,1 



21 



/.log. 11-1-0413927. 



But log. 10 - 1. 



To seven places of decimals, 



which is the logarithm you will find registered in the 

 table. From the formula 



lo^-^S + i'^' 



3" (.q+pJ^ 

 Another very convenient formula can be derived. 



j 

 Now log. x ,_ 1 . 



2 log. x log. (x 1) log. (x + 1). 



20. To explain the use of this last Series. 



The use of this formula, which converges very 

 rapidly, gives us any logarithm in terms of the two that 

 precede it. It will be observed in this formula that 



<r in i i\ and all the terms following it can be 

 omitted, provided 



IF (2x J I) 



and , 

 and . 



000000 l 



) 8 < 0000002 7- 

 <'003. 



2x _ 1 



Or if 2x 1>334. 

 .- . if x- > 169. 

 . . if x > 13. 



Hence, if we employ this formula to calculate logarithms, 

 we have, for all numbers greater than 13 



log. (x + 1) - 2 log. x log. (x 1) g^TTT 

 Agrxin, suppose x > 10000. Then 



- < -oooooooooi 



> -047C55089 X '868688962 

 " -041392674 



must be at least many hundreds for -y to differ sensibly 



and 



2*f < . 00000001. 



Hence, the formula finally reduces itself to 

 log. (x + 1) - 21og. x-log. (x- 1)- X T- (IX.) 



Or log. (x + 1) log. x - log. x log. (x 1) . 



M <2 3n & 

 ^ ?. < - i- occ - 



-r ~~ 7 ^i \> ?. i 



x 1 (x-f-n)* x ( x z' 



. . if x > 10000, in which case /J- > .00000001. 

 Hence, if we only calculate to 7 place* of decimals, n 



from , ' TJ' Now, the formula would give us 

 log. (x + n + 1) log. (x 4- n) - log. (x + n) 



log.(x + n-l)- 



(x 



which, unless n is several hundreds, we have seen is 

 practically the same as 



log. (x + n) log. (x + n 1) = log. (x + n 1) 



-log. (x + n-2)- - 

 So that log. (x + 1) log. x = log. x log. (x 1) 



log. (x + 2) log. x - log. (x + 1) log. x . . 

 log. (x + 3) log. (x + 2) = log. (x + 2) 



log.(x+l) . 



log. (x+n) log. (x + n 1) = log. (x + n 1) 

 log. (x + n 2) 



.'. adding together 



log. (x + n) log. x = log. (x + n 1) log. (x 1) 

 _ j_ 



. . log. (x + n) log. (x + n 1) = log. x 

 log. (x 1) n. 



Now, if n be sufficiently small for?i_- to be < -0000001, 



it is clear that we may omit n. -^- so long as this is the 



case, and hence the differences between the successive 

 logarithms will continue the same within that limit. 

 For instance, we can show by formula (VIII.) that 



log. (10000) = 4-00000000. 



log. (10001) = 4-00004342945." 



.-. log. (10001) log. 10000= -00004343. 



(10000) ! 



I 



= -0000000042934. 

 r < -0000001. 



and 



Hence n (100000)1 



until n = 20. So that the logarithms of 10001, 10002, 

 10003 .... 10020. can be found the one from the one 

 before it by merely adding .0000434294. 



We shall then have to calculate log. (10020) and log. 

 (10021) from the original fonnuh> (VIII.) and find how 

 far we can use the difference between these for deducing 

 log. (10022) by log. (10023). <bc. After some 20 or 30 

 logarithms are thus found by simple addition, a fresh 

 calculation will become necessary : by proceeding in this 

 manner, without any exorbitant labour, a table which 

 gives the logarithm for every numberf rom 1 0000 up to 



