 Pathwise uniqueness for a multiplicative martingale equation

The Itô formula gives $d\log Z=dM-\frac{1}{2}d[M]$ for a strictly positive solution. Therefore equal initial values give indistinguishable solutions. Conversely the displayed exponential satisfies the equation. This is uniqueness for a fixed continuous local martingale driver and does not require the exponential to be a true martingale.