Verified answer

No because as much as you try you can't write it as f(x | θ) = h(x) exp(η(θ) T(x) - A(θ)) -- the density function for a single-parameter exponential family. The closest we can get is f(x | θ) = 2 exp(log(x θ) - log(1 2θ)) but at this point log(x θ) cannot be factored (decoupled) as a function of θ only times a function of x only.

Note on terminology: When we do have an exponential family's density function written out, T(x) is known as "sufficient statistic," η(θ) is called "natural parameter," h(x) is termed "base measure" and A(θ) the "log partition."