3S Exponential and Logarithmic Theorems. 



'L.{p-\-q), denote the hyperbolic logarithms of N and ^+g. 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 



they become L.(l+a)=a—-o-+-o--&:'C., (9), N'' = l-fa;L.N-f 



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

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



by (7) log.(l + 6)=log.e|^6--^4-3--&c.j=l, and by (9) we 



have 6 — -q'+q'— &c. = L.(I+6), .*. by substitution log.exL. 



1 



(1-f 6)=::lj or log.e=j^|-rTT, and if we put l-{-6=A, log.e= 



1 



J— X, •'• substituting this value of log.e in (7), it becomes log. 



I ( a^ a^ \ ^ 1 



(l+a)=T— -rla— -^-}--o' — &c. ), (11), where j—r 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 ; 



a^ a^ 

 .'.since by (9)a--2 +"3-— &c.=L.(l+a), (11) becomes log. 



(l+a)=mla— -o'+'o' — 2r4-&c. ) = mL.(l+a), (12), so that 



to find log.(l4-ff) we must multiply the hyperbolic logarithm of 

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



1 



we have ^=rTQ> ^i^d 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" «=* a" \ / 



1«-1-"o'+'q'+'T"1''^^V =*w.L.(1 — a), or I since by the nature of 



logarithms log.(l-a)= -log. (j^^^j j, log. (yz^) =m [a-\- ^ 



a^ a' \ ^ / 1 \ 



+ -o-+-x+&'<^-) =*^L Ij-^— I, (13), which added to (12), gives 



/l+a\ ( a^ a' \ (l+ct\ 



log.\i:i^j=3wi|^a+-3-4-5-+&c.l=mH^j3^j, (14); if we 



