I think this is mostly unavoidable: cmath.log(z, 10) is a compound operation that does the equivalent of cmath.log(z) / cmath.log(10), while cmath.log10 is doing everything at once.

If anything, this is a problem in how complex division is done: it's arguable that division by a complex number with zero imaginary part should be special-cased here.

>>> inf = float('inf')
>>> z = complex(-0.0, -inf)
>>> cmath.log(10)
(2.302585092994046+0j)
>>> cmath.log(z)
(inf-1.5707963267948966j)
>>> cmath.log(z) / cmath.log(10)
(inf+nanj)

A simpler example just involving division:

>>> complex(2.0, inf) / 2.0   # expect 1 + infj.
(nan+infj)