Functions - finding the domain Question:

Consider the function: $$f(x) = \log(2x + 1) - \log(x - 3)$$
  What will be the domain of this function?

I used two approaches to solve this question. Both approaches got me different answers. Consider that we do not merge the two $log$ together. As we know that the value inside a $\log$ should be greater than zero:
$$2x + 1 \gt 0$$ 
$$x - 3 \gt 0$$
From these two inequalities, we get that $x \gt 3$
Now consider that we merge together the two $\log$:
$$\log(2x + 1) - \log(x - 3) = \log(\frac{2x + 1}{x - 3})$$
We now know that the value inside the log has to be greater than zero. This can be in two conditions. Either both the numerator and denominator are positive, or both are negative.
Thus we get:
$$x \in (-\infty, -\frac{1}{2}) \cup (3, \infty)$$
This is perfectly valid when we simplify the expression. However, if we don't simplify, the interval $(-\infty, -\frac{1}{2})$ becomes invalid for $\log(x-3)$ as the value inside is negative. Which answer is correct?

The rule $\log a-\log b=\log\frac ab$ holds only on the condition that $\log a$ and $\log b$ exist. So when you're applying that rule you're implicitly assuming $x>3$, and this assumption has to be combined with the condition for $\frac{2x+1}{x-3}$ to have a logarithm.
