I am trying to move the $x$ on the right over to the left side of this equation to solve for $x$:
$$x = \large e^\frac{{{{ \Large(z / x - 1 - 0.2029)}}}} {{\large {-0.022}}}$$
I am basically trying to get the reverse of equations I have to get $z$:
$$y = -0.022 *\ln(x) + 0.2029 + 1$$ $$z = xy$$
I am have created the above equations, but having trouble figuring out how to move X out of the base e's exponent. Any help would be appreciated. Thanks.
You will not be able to do that: there is no way to bring $x$ "out of the base $e$'s exponent" in the simple equation $x=e^{z/x}$. In fact the solution, which is $x=z/W(z)$, involves the Lambert $W$ function.
Your equation is of the form $$ x = e^{a(z/x+b)}. $$ The solution is $$ x = \frac{az}{W(ae^{-ab}z)} . $$