If $1, \log _{10}\left(4^x-2\right)$ and $\log _{10}\left(4^x+\frac{18}{5}\right)$ are in arithmetic progression for a real number $x$, then the value of the determinant $\left|\begin{array}{ccc}2\left(x-\frac{1}{2}\right) & x-1 & x^2 \\ 1 & 0 & x \\ x & 1 & 0\end{array}\right|$ is equal to :
