Is logarithms required?

Algebra Level 3

If $$2^{x}=4^{y}=8^{z}$$ and $$xyz= 288$$, with $$\frac{1}{2x} + \frac{1}{4y}+ \frac{1}{8z}= \frac{m}{n}$$ for coprime positive integers $$m$$ and $$n$$, find $$m+n$$.

×