Natural logarithm with absolute value: Can I cancel the absolute value?

Question

I was calculating basic rational integrals and came up with this kind of problem. I have this expression:

$$2\ln|x|$$

I can re-write it down like that:

$$\ln{x^2}$$

and thus cancel the modulus.

The question is, what about $\frac{1}{2}\ln|x|$?

Should I write it down like this:

$$\ln{\sqrt{x}}$$ or like this: $$\ln{|\sqrt{x}|}\,?$$

Answer 1

As $\ln\sqrt{\lvert x\rvert}$. Otherwise, think about what would happen if $x$ was equal to, say, $-1$.

Answer 2

There is no reason to remove the absolute value.

$$\frac12\ln|x|=f(|x|)$$

which doesn't fit with your proposals.

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This works with the square, because

$$x^2=|x|^2=f(|x|).$$