Utilizing Compact Logarithmic Tables. 223 



annexed to the whole number x, just as if the number were 

 written 3141-592 &c. ■ it is plain that the right index can be 

 supplied afterwards. Then the Tables give 



Log — x. Log a?. 



x 



61407164 3141 3*49706 79364 



log n 9-7728009 



1 Mn 



2 x 



5-9135173 



-410 



5-9134763 log 8 19363 



3-49714 98727 



Here, adding to 6*140 &c. (which is found in the Tables) the 

 logarithm of n, we obtain the logarithm 5*9135173, which being 

 sought for in Hutton's Tables (p. 149), the first few figures are 

 8195, one-half to the nearest digit (the index 5 indicating its posi- 

 tion) being subtracted leaves 5*9134763, which is the logarithm 

 of -00008 19363 ; this added to log 3141 gives the exact loga- 

 rithm required* 



This is very simple and easy, the only puzzle being the index 

 and proper position of the figures to those who are not much in 

 the habit of using these large numbers ; but two examples at the 

 bottom of each page of the Tables causes this difficulty to be soon 

 surmounted. It may be remarked that when n is negative, the 

 half is to be added, not subtracted. 



The reverse operation is equally simple and effective. Let 



, x + n 



then 



n _ a 1 « 2 la 3 



x~U + 2W + 6W + ' 



and 



, n , a \ le 2 _ 1 a 4 



log- =log H + r + M jp + 0- ^jp + &C 



whence 



log n— log x + o a + l°g tT? + log a ver y nearly. 



* If d be the number of digits, then the formula is correct to 3rf+ 1 . In 



M 



point of fact the log — is never copied out from the Tables, but a slip of 



paper with logra on its upper edge is brought up to it, and the final number 

 only is written in under log x. 



