24^ LOGARITHMS, 



that the common logarithm of any number lo" or N is ft 

 the index of that power of lo, which is equal to the said 

 number. Thus, loo, being the second power of lo, will 

 have 2 for its logarithm j and looo, being the third power 

 of lo, will have 3 for its logarithm : hence also, if 50 be 

 ---jqI'6 9 8 9 7^ then is i "69897 the common logarithm of 

 50. And, in general, the following decuple series of 

 terms, 



viz. 10*, loS 10% 10% 10*, lo""', 10"*, 10""*, IQ-S 



or loooo, 1000, 100, 10, I, 'I, '01, '001, '0001, 



have 4, 3,. 2, I, o, — i, —2, — 3, —4, 



for their logarithms, respectively. And from this scale of 

 numbers and logarithms, the same properties easily follow, 

 as before mentioned. 



PROBLEM. 



To compute the logarithm to any of the natural numherSy i, 2, 

 3, 4, 5, ^c, 



RULE. 



Let b be the number, whose logarithm is required to be 

 found ; and a the number next less than ^, so that b — a'zz 

 I, the logarithm of a being known ; and let s denote the 

 sum of the two numbers ^r-f-^. Then 



1. Divide the constant decimal '8685889638, &c. by /, 

 and reserve the quotient ; divide the reserved quotient by 

 the square of s, and reserve this quotient 5 divide this last 

 quotient also by the square of j, and again reserve the quo- 

 tient ; and thus proceed, continually dividing the last 

 quotient by the square of j", as long as division can be 

 made. 



2. Then write these quotients orderly under one anoth- 

 er, the first uppermost, and divide them respectively by 



the 



