Note that there are eleven equations, but fifteen variables ($z$, $e$, $r$, $o$, $n$, $t$, $w$, $h$, $f$, $u$, $i$, $v$, $s$, $x$, and $g$). So there won't be a unique solution - at best, you could hope to find eleven of them in terms of the other four.
But taking the logarithm of each equation will make this doable: for example, $\ln{o * n * e} = \ln 1$ tells you that $\ln{o} + \ln{n} + \ln{e} = 0$. Then you can solve the system of linear equations for the logs of the variables; once you've done that, undo the logarithm to get the value of the variables.