You could certainly choose to go that route. In that case, the resulting equation would be quadratic as opposed to linear and the result would be $x = \displaystyle \frac{x^2}{4}$, which would also give you a unique solution to the value of $x$, which is $4$.
For the second statement, why didn’t we input the value of x in the exponent on left side? Can we be selective in terms of putting the value of x?
I mean exponent on right side of the equation
You could certainly choose to go that route. In that case, the resulting equation would be quadratic as opposed to linear and the result would be $x = \displaystyle \frac{x^2}{4}$, which would also give you a unique solution to the value of $x$, which is $4$.