Question

If $\log _2\left(2 x^2\right)+x^{\log _x\left(1+\log _2 x\right)} \log _2 x+\frac{1}{2}\left(\log _4 x^4\right)^2+x^{3 \log _x\left(\log _2 x\right)}=27$, then find the value of x.

Answer 1

Let $\log _2 x=t$, So we get
$(1+2 t)+(1+t) t+2 t^2+t^3=27 \Rightarrow t^3+3 t^2+3 t+1=27$
$\Rightarrow (1+t)^3=27 \Rightarrow 1+t=3 $
$\Rightarrow t=2 \Rightarrow \log _2 x=2, \text { so } x=4$