Power on absolute value expressions

Question

How to deal with absolute value, when raised to odd number?

Like here: $$\left|\log_2\left(\frac{x}{2}\right)\right|^3 + |\log_2 (2x)|^3 = 28$$

Answer 1

We have $\log_2\left(\frac{x}{2}\right)=\log_2 x-1\; \text{and} \; \log_2\left(2x\right)=\log_2 x+1.$

Set $a=\log_2x.$ The equation rewrites $$\left|a-1\right|^3 + |a+1|^3 = 28.$$ Solve separately for $\; a<-1,\; -1\leq a \leq 1, \; a>1.$ We get $a=-2$ or $a=2.$
The solutions of the initial equation $x\in\{{1\over 4}, 4\}.$