Logarithmic rules

Question

So we know the formula $\ln(ab) = \ln(a) + \ln(b)$, but say I choose $a=-2$ and $b=-1$ we have $\ln(2) = \ln(-1) + \ln(-2)$ which is wrong as $\ln(x)$ only valid for $x>0$. What's wrong with this?

Answer 1

Every rule is comprised of two parts: its domain and its application. The rule with logarithms is:

If $a$ and $b$ are positive real numbers, then $\ln ab = \ln a + \ln b$.

So you cannot choose $a=-2$, and everything you do after you choose $a=-2$ is irrelevant.

---

Also, as an aside, I advise you try to separate, in your mind, the concept of a wrong statement from the concept of an undefined statement.

For example, $1 + 1 = 3$ is wrong, but $\frac00=1$ is not so much wrong as it is not really an equation, because the left hand side of the "equation" is not defined.