﻿38 Exponential and Logarithmic Theorems. 



Li.(p + q), denote the hyperbolic logarithms of N and p+q. If 

 the logarithms in (7) and (8) are supposed to be taken to the 

 base e, (or are hyperbolic,) then since e is the base, Le=l, and 



a 2 a 



they become L.(l+a)=a—-o-+^--&c. ; (9), N I = l+a;L.N+ 



(rrL,N)' ( g L,N)» 



-L2 _ + T2X- +ifcc - CIO). 



If we suppose that the logarithms in (7) are taken to the base 

 1 + 6, we shall have, since 1+6 is the base, log.(l + 6) = l, and 



b 2 b 3 



by (7) log.(l + 6)=log.e^-^+y-&c.J=l, and by (9) we 



b 2 b 3 

 have b — tt + o~ — &c. = L.(l+6), .\ by substitution log.eXL. 



1 



(1+6) — 1, or l°g-* = T^fXTy an( l if we put 1+6=A, log.e 



1 



y— r, /. substituting this value of log.e in (7), it becomes log, 



1 / a 2 a* \ 1 



(l+a)=j— rl^ — "o"+ o" — &c. ), (11), where j— j- is called the 



modulus of the system of logarithms when A is taken for base ; 



we shall denote the modulus by m, and m will equal unity divided 

 by the hyperbolic logarithm of the number that is taken for base j 



a 2 a 3 



.•.since by (9)a- ?r +~o~— &c.=L.(l+a), (11) becomes log. 



/ a 2 a 3 a 4 \ 



(l+a)=mla— 0-+-0 — j-+&c.) = mL.(l+a), (12), so that 



to find log.(l+a) we must multiply the hyperbolic logarithm of 

 (1+a) by the modulus j -. in the common system where A=10, 



1 



we have m=j— tt;, and if we multiply the hyperbolic logarithm 



of any number by this value of m, we shall get the common 

 logarithm of the number. 



If we change the sign of a in (12), we get log.(l-a)= — m 



a 2 a 3 a* \ ( . 



a + 2"+ 7T+-X+&C. J =mL(l-a), or I since by the nature of 



logarithms log.(l-a)==~log.(:^)), log. (;fzj =m[a+ a ^ 



a 3 a* \ I 1 \ 



+ T+T"t" < k c 7 =m L ( j— — L (13), which added to (12), gives 



/l+a\ a 2 a 5 \ /1+<M 



log^iZ^J= 2 ^l«+^ + y+&c.j=mL^^^ (14); ^ we 



