This is hard to solve because there are no elementary manipulations that will convert this into a form you can use. That is, solving for $n$:
$2^n = n^8$
is not a problem that is reasonable to be solved in a pre-calc or calculus type class.
In fact, there is a symbolic solution to this, but it's sorta cheating. Suppose you have a function $W(x)$ such that
$y = W(x) e^{W(x)}$
(This is called the Lambert W Function )
then you can use $W$, with suitable algebraic manipulation, to help solve weird relations like yours (um...Arturo did this for you so thankfully I won't have to bother now).
Then you can use numerical properties of $W$ to get actual values.
The cheating part is because you just assume there's a solution function of a certain form $W$ with its properties) and manipulate your original problem to reduce to using $W$ (as Arturo did). And then, because other people have done lots of numerical analysis on $W$, you can (using tables or a computer algebra system like Maple or Mathmatica) get a numerical answer.
But if all you care about is a symbolic answer, you can stop at Arturo's.
Which is all to say, you shouldn't be ashamed at all, the equation you have is not solvable with the tools you have at your disposal.