Basic doubt about exponents. I have a doubt regarding this:
$\begin{aligned}  \ln y &=\ln \left(x^{x^{x}}\right) \\ &=x \ln \left(x^{x}\right) \\ &=x \cdot x \cdot \ln x \\ &=x^{2} \cdot \ln x \\ &=\ln \left(x^{x^{2}}\right) \neq \ln \left(x^{x^{x}}\right) \end{aligned}$
I KNOW that I did it wrong ( in step 3) but can any body explain me what exactly I did wrong and point out the exact fundamental  understanding that I lack.
Please try to be extremely specific while explaining the error.
If u want more info on the question, feel free to comment.

Your error was due to incorrect association of the exponents:
$$ x^{x^x} = x^\left(x^x\right) \neq \left(x^x\right)^x. $$
As you know,
$$ x \ln(u) = \ln\left(u^x\right). $$
Observe that
$$ x \ln \left(x^x\right) =   \ln \left(\left(x^x\right)^x\right)
 \neq \ln \left(x^{\left(x^x\right)}\right) = \ln\left(x^{x^x}\right). $$
Also observe that
$$ \left(x^x\right)^x = x^\left(x^2\right). $$
And note that you are inconsistent in the way you associate exponentiation, because you treated $x^{x^x}$ as $\left(x^x\right)^x$
but you treated $x^{x^2}$ as $x^\left(x^2\right).$
