Integral of the Logarithm of Dirac Delta function?

\int ln(\delta (x)) dx

over infinity. I understand the properties of a Dirac delta, it is zero everywhere except when its argument is equal to zero, where it is infinity. The integral of a dirac delta function over all values of it argument is equal to one. But what about when it's logged? I'm having a hard time grasping that, from my notes it SHOULD be equal to zero. I'm guessing because we're left with