The Definition of the Absolute Value The Absolute Value can be defined in many ways, but these are the two most common :
1. As a Piecewise Function
$$
|x|=
\begin{cases}
-x&\text{if } x < 0\\
x&\text{if } x\geq 0
\end{cases}
$$
2. As The Principle Square Root of a Square
$$|x| = \sqrt{x^2}$$

In the second definition that I've included here, what stops us from doing the following, and reaching a contradiction?
$$|x| = (x^{2})^{\frac{1}{2}} = x \ \ \ \ \ \ \ \ \ \  \ \text{Contradiction}$$
Likewise, if I have $f(x) = \ln(|x|)$, what is the reason why the following contradiction can't be reached :
$$f(x) = \ln(|x|)$$
$$\implies f(x)=\ln(\sqrt{x^2})$$
$$\implies f(x) = \ln[(x^2)^{\frac{1}{2}}]$$
$$\implies f(x) = \frac{1}{2}\ln(x^2)$$
$$\implies f(x) =  \frac{1}{2} \cdot 2 \ \ln(x)$$
$$\implies f(x) = \ln(x) \ \ \ \ \ \ \ \ \ \  \ \text{Contradiction}$$
 A: $\sqrt{x^2} = (x^2)^{\frac{1}{2}}=x$, only when $x>0$, otherwise it equals
$-x$.
This is because for any $x>0$, $|x|=x$ and for any $x<0$, $
|x|=-x$
So, no contradiction is reached.
A: Because $(x^2)^{1/2}$ is not in general equal to $x$. The squaring operation lost the information about the sign of $x$.
With the log example, it's because you need to pick a branch.
A: What stops us from saying $|x|= (x^2)^{1/2}= x$ is that the second equality simply is NOT true!  $a^{1/2}$ is defined as "the positive number $x$ such that $x^2= a$".  In particular, if $x= -2$, then $x^2= (-2)^2= 4$ and then $((-2)^2)^{1/2}= 4^{1/2}= 2$, not $-2$.
A: First off, I want to thank everyone for their answers here, but I found this to be the most general answer after thinking about this question for a while. 
$$(a^{n})^{\frac{1}{n}} = a \ \ ,\ \text{if} \ \  n \ \ \text{is odd}$$
$$(a^{n})^{\frac{1}{n}} = |a|\ \ ,\ \text{if} \ \  n \ \ \text{is even}$$
Stated a bit more formally :
$$(a^{2n+1})^{\frac{1}{2n+1}} = a \ \ ,\ \forall n\in\mathbb{Z}$$
$$(a^{2n})^{\frac{1}{2n}} = |a|\ \ ,\ \forall n\in\mathbb{Z}$$

Parts of this answer and the intuition developed behind it have come from Paul's Online Notes, so full credit must go to that site.
