Matrices and Determinants: JEE Main 2021 Mathematics question

If 1, log₁₀(4^x $-$ 2) and log₁₀$\left( {{4^x} + {{18} \over 5}} \right)$ are in arithmetic progression for a real number x, then the value of the determinant $\left| {\matrix{ {2\left( {x - {1 \over 2}} \right)} & {x - 1} & {{x^2}} \cr 1 & 0 & x \cr x & 1 & 0 \cr } } \right|$ is equal to :