With no base argument log of complex zero is
> log(0i, )
Ideally, the same answer should be given in both cases.
By the definition log(x,b) = log(x)/log(b)
x <- log(0i) # -Inf+0i
x/log(exp(1)+0i) # -Inf+NaNi
x/log(exp(1)) # same
x/1 # same
That's odd, but as far as I can see is being done by the platform's C99 complex arithmetic and not R.
I should have said 'odd but correct'. Consider
x <- -Inf+0i
y <- 1
Then the imaginary part of the result is
x.r*y.i + x.i*y.r = -Inf*0 + 0*1 = NaN.