$\ln ab - \ln |b| = $
Options:
(a) $\ln a$ ;(b) $\ln|a|$ ;(c) $-\ln a$ ;(d) none of these.
My attempt: $\ln a + \ln b - \ln |b| = \ln a + \ln {\frac{b}{|b|}}$.
Now, $\frac{b}{|b|}=\pm1$, but it can't be $-1$ for log to be defined. So, it means $b$ is positive. So, for $\ln ab$ to be defined, $a$ should be positive too. So, the answer should be option (a). Or, at max both option (a) and (b). But the answer has been given as (b). What's your take?
The given answer (b) is correct.
If $\ln ab$ is defined then $ab>0$ so $|ab|=ab.$
Thus $\ln ab-\ln |b|=\ln |ab|-\ln|b|=\ln|a||b|-\ln|b|=\ln|a|+\ln|b|-\ln|b|=\ln|a|.$
Note that it is possible for $a$ and $b$ to be both negative or both positive here.