Question

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

Answer 1

Answer: 4

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