Does $\left|\log^2{x}\right|=\log^2x$ hold true?

Question

I have a question about absolute values. Does the following hold true? $$\left|\log^2{x}\right|=\log^2x$$

In one problem my textbook removes the absolute from $\left|\log^{2/3}x\right|$.

In theory yes, because the absolute value of $|x|=x$ if x is positive or equal to $0$. Sorry for the dumb question and thanks.

Answer 1

Your question has nothing to do with logarithms. For every real function $f$ it is true that $(\forall x\in D_f):\bigl\lvert f^2(x)\bigr\rvert=f^2(x)$ simply because $f^2(x)\geqslant0$.

Answer 2

Yes of course for $x>0$ since $\log^2 x \ge 0$ we have

$$\left|\log^2{x}\right|=\log^2x$$

Answer 3

Sometime ago I had the same frustration. Generally, we do not accept no integers powers of negative numbers such as $(-1)^{3/2}$ or $(-1)^{\pi}$ e.t.c, but in case where we have a rational number with odd denominator such as $2/3$, $6/5$ etc, it is possible to write $(-1)^{2/3}$ which of course is equal to $\left((-1)^2\right)^{1/3}=\left((-1)^{1/3}\right)^2$ .