Approximating $ | \ln(x)| $ by $-\ln(x)$ near 0

Question

I am reading a text on the dominated convergence theorem. As I am going through the examples, I can see that some of them use the fact that, when $x$ is close to $0$: $$ |\ln(x)| \approx -\ln(x) \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,[x \approx 0]$$ I won't quite see where this come from. A am familiar with the usual Taylor developments such as those of $\ln(1-x)$ but still, I don't see if it's of any help to understand the above approximation. Any insight would be appreciated, thanks.

Answer 1

Recall that by definition

• $|x|=x\quad$ for $x\ge 0$
• $|x|=-x\,$ for $x< 0$

and then, since $\ln x <0$ for $0<x<1$, we have that $$|\ln x|=-\ln x=\ln \frac1x$$

Answer 2

The "$\approx$" should rather be a "$=$", because they are exactly the same quantity for all $x\in (0,1)$.