The numbers \(-16,2^{x+3}-2^{2 x-1}-16,2^{2 x-1}+16\) are in an arithmetic progression. If \(x\) is an integer, then \(x\) equals

The integer value is $3$. Twice the middle term equals the sum of the outer terms, which for $t=2^x$ simplifies to $3t^2-32t+64=0$. The roots are $8$ and $\frac83$, and only $8=2^3$ is an integer power of $2$. A value of $8$ reports $t$ rather than its base-two logarithm.