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

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: $$\ln{|\sqrt{x}|}\,?$$

• You should write $\ln{\sqrt{|x|}}$ – Gabriele Cassese Mar 22 at 18:59

As $$\ln\sqrt{\lvert x\rvert}$$. Otherwise, think about what would happen if $$x$$ was equal to, say, $$-1$$.
$$\frac12\ln|x|=f(|x|)$$
$$x^2=|x|^2=f(|x|).$$