Domain of truth

We know that, ${\color{#20A900}{\log(xyz) =\log (x)+\log (y) + \log (z)}}$

But it's not true for, ${\color{#D61F06}{\log(xyz) = \log (x+y+z)}}$

However, for some values of $x,y$ and $z$ the false property above is true.

If, ${-10\leq x,y,z\leq10}$ and $x, y, z$ are integers, then find the total number of ordered triples ${(x,y,z)}$ for which the equation above is true.

Note: The domain of the $\log$ function is positive reals.