It doesn't get much more simple
$\displaystyle x = 2^{\log_23+\log_28}$
$\displaystyle x = 2^{\log_23+\log_22^3}$
$\displaystyle x = 2^{\log_23+3\log_22}$
$\displaystyle x = 2^{\log_23+3\times 1}$
$\displaystyle x = 2^{\log_23+3}$
$\displaystyle x = 2^3\times 2^{\log_23}$
$\displaystyle x = 8\times 2^{\log_23}$
This is the most fundamental truth about exponentials and logarithms that you really need to understand: one is the inverse of the other. That is, logarithms "undo" exponents, and exponents "undo" logarithms.
$\displaystyle a = \log_b c \iff c = b^a$ for any $\displaystyle a$, and any $\displaystyle b, c$ positive. That's how logarithms are defined.
So $\displaystyle 2^{\log_2 x} = \log_2 (2^x) = x$.