$\ln ab - \ln |b|$

Question

$\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?

$a$ and $b$ could both be negative, and $\ln ab - \ln|b|$ would be defined — J. W. Tanner, Aug 06 '19 at 16:36
Since it is given that $\ln(ab)$ is defined, hence $ab >0$ it follows that $\ln(ab) - \ln |b|=\ln |ab| - \ln |b|= \ln |a|$ — Conrad, Aug 06 '19 at 16:39
So, then can we not deduce that both $a$ and $b$ should be positive? — aarbee, Aug 06 '19 at 16:40

score 1 · Accepted Answer · answered Aug 06 '19 at 17:06

1

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.

answered Aug 06 '19 at 17:06

J. W. Tanner

60,406

$\ln ab - \ln |b|$

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