When solving log equations such as: log(x) + log(x - 3) = 1 we find the "potential solutions" to be x = 5,-2. But only 5 is a solution because we cant generate logs of negative numbers. What really gets me is that I'm told I in order to check my potential solutions I have to plug them back into the original equation because by combing the logs to solve the equation such as in log(x / (x -3)) we have "CHANGED" the problem and it's no longer the same and the solutions in effect don't apply after rewriting it. But why is that? Are not the two expression equal?
How can we say log(x) + log(x - 3) = log(x / (x - 3)) but in the same breath call the right side of that equation changed and different and in fact have different solution sets ?
If $x$ is such that $\log(x)$ and $\log(x-3)$ are defined, then yes, $$\log(x) + \log(x - 3) = \log\frac x {x - 3}$$ However, the reverse is not true. I can choose for example $x=-10$, then $$\frac x{x-3}=\frac{10}{13}$$ I can take the logarithm of this expression, even if $\log(-10)$ and $\log(-13)$ are not defined.