412 Professor HAmILTon on Conjugate Functions, 
and that it increases constantly and continuously from positive states indefinitely near 
to 0 to positive states indefinitely far from 0, while a increases or advances constantly, 
and continuously, and indefinitely in the progression from contra-positive to positive ; 
so that, for every positive number /, there is some determined number a which satis- 
fies the condition 
B=r (@), (120.) 
and which may be thus denoted, 
a=F-' (). (121.) 
It may also be easily proved that we have always the relations, 
F (a) =e", F—'(B) =log,. B, (122.3 
if we put, for abridgement, 
F (1) =e, (123.) 
and employ the notation of powers and logarithms explained in the Preliminary 
Essay. A power b when considered as depending on its exponent, is called an 
exponential function thereof; its most general and essential properties are those 
expressed by the formule, 
rele sweaty (Sti (124.3 
of which the first is independent of the base }, while the second specifies that base ; 
and since, by (113.), the function-couple F (a, @,) satisfies the analogous condition, 
F (@, a2) x F (b,, 6 =P ((, 2) + (A, 0.) JHE (a, +5, a. + dy), (125.) 
(whatever numbers a, a, b, b, may be,) we may say by analogy that this function-couple 
F (a), a) is an exponential function-couple, and that its base-couple is 
1 (HS MEG; O) 2 (126.) 
and because the exponent a of a power &, when considered as depending on that 
power, is called a logarithmic function thereof, we may say by analogy that the 
couple (a), a.) is a logarithmic function, or function-couple, of the couple F(a, @,), 
and may denote it thus, 
(a, a.) =Fa'G, 4), if ©, 4)=r@, a,). (127.) 
