What's wrong with this logarithm calculation? We know that $\displaystyle\log_a(xy)=\log_ax+\log_ay$. 

Consider the following:


  $$\displaystyle\ln(1)=\displaystyle\ln((-1)\times(-1))=\displaystyle\ln(-1)+\displaystyle\ln(-1)
$$


$\displaystyle\ln(1) $ is a completely valid statement, but I'm not sure if $\displaystyle\ln(-1)+\displaystyle\ln(-1) $ is.

$\displaystyle\ln(-1) $ doesn't exist, but $\displaystyle\ln((-1)\times(-1)) $ does, and if I plug $\displaystyle\ln((-1)\times(-1)) $ into my calculator it gives a 0 as the answer (which is correct), but if I plug in $\displaystyle\ln(-1)+\displaystyle\ln(-1) $ it says that there is a domain error (which there is). So my question is, what is wrong with the highlighted equation?
$$\displaystyle\log_a(xy)=\log_ax+\log_ay$$


The above identity holds only for $x,y&gt;0$ and $a\not =1$, $a&gt;0$.
Basically, once you go to taking the log of negative numbers, and therefore getting complex results, you are into the realm of multi-valued functions. Cf. $arcsin(\theta)$ etc.
Euler's identity says $exp(i\pi) = -1$ But also, for example, $exp(-i\pi) = -1$. So by taking logs of both sides, we get $log(-1) = (2n - 1)i\pi$ for an arbitrary integer value of n, and it is effectively by convention that we take $n = 0$ to get the principal value of the log function..

Then the effect of writing $log(1) = log(-1) + log(-1)$ simply amounts to:
$$0 = (2m - 1)i\pi + (2n - 1)i\pi$$which reduces to choosing m, n such that $m + n = 1$.

This is the same order of paradox as saying $\sqrt4 = 2$ and $\sqrt4 = -2$. By convention we take the principal (positive) value throughout; but there are circumstances where we have to break the convention to get a sensible result.