If the fourth term in the expansion of x + x log 2 x 7 is 4480 , then the value of x where x ∈ N is…

Given ${\left(x+x^{{\log }_{2}x}\right)}^7.$

The general term in the expansion of ${\left(x+x^{{\log }_{2}x}\right)}^7$ is ${}^{7}C_r{x}^r{\left(x^{{\log }_{2}x}\right)}^{7-r}.$

Then, $T_4=$${}^{7}C_3{x}^4{\left(x^{{\log }_{2}x}\right)}^3=4480$

$\Rightarrow \frac{7·6·5}{3·2·1}{x}^{4+{\log }_{2}x}=4480$

$\Rightarrow x^{\left(4+3{\log }_{2}x\right)}=128$

$\Rightarrow x^{\left(4+3{\log }_{2}x\right)}=2^7$

Now, from the option only at $x=2$, $\mathrm{LHS}=\mathrm{RHS}.$