197] 
NOTE ON THE THEORY OF LOGARITHMS. 
213 
It thus appears that when the real parts «, x', xx — yy’ are all three of them 
positive, or any two of them positive and the third negative, E is equal to zero, or 
the logarithm of the product is equal to the sum of the logarithms of the factors; 
but that if the real parts are one of them positive and the other two of them 
negative, then if a certain relation between the real and imaginary parts is satisfied, 
but not otherwise, the property holds; and if the real parts are all three of them 
negative, the property does not hold in any case. 
The preceding results do not apply to the case where any one of the arguments 
x + yi, x + y'i, (x + yi) («' + y'i) is real and negative, for no definition applicable to 
such case has been given of a logarithm. If, however, we assume as a definition 
that the logarithm of a negative real quantity is equal to the logarithm of the 
corresponding positive quantity, then in the case, x = —, y = 0, we have 
log x + log (x' + y'i) — log x {pc' + y'i) = 67ri, e = ± 1 = y'; 
an equation which is, in fact, equivalent to 
log (x' + y'i) — log [— («' + y'i)] = eiri, e — ± 1 = y'; 
and in the case xy + xy = 0, xx' — yy' = —, which implies y = y', then 
log (x + yi) + log (x' + y'i) — log (x + yi) (x + y'i) = 7ri, e = ± 1 = y or y'; 
an equation which is in fact equivalent to 
log (x + yi) + log (— X + yi) — log (x 2 + y 2 ) = 67ri, 6 = + 1 = y. 
The case where both of the arguments x + yi, x' + y'i are real and negative, i.e. 
x = — , y = 0, x' = —, y' = 0 gives of course log x + log cc — log xcc = 0, the logarithms of 
the negative real quantities x, x' being by the definition the same as the logarithms 
of the corresponding positive quantities. It should, however, be remarked that the 
definition, (x= — ), logx — log(— x) not only gives for log« a different value from that 
which would be obtained from the general definition of a logarithm, by considering 
log« as the limit of log {x + yi, y = + , or of log {x + yi), y = —, but gives also a value, 
which, for the particular case in question, contradicts the fundamental equation 
e Xosx = x. It is therefore, I think, better not to establish any definition for the logarithm 
of a negative real quantity «, but to say that such logarithm is absolutely indeter 
minate and indeterminable, except in the case where, from the nature of the question, 
x is considered as the limit of «+ yi, y positive, or of « + yi, y negative. 
2, Stone Buildings, March 15, 1856.