I am interested in computing the first group cohomology $H^1(\mathbf{R}^\ast, \mathbf{R})$, where $\mathbf{R}^\ast$ is acting on $\mathbf{R}$ by multiplication (here $\mathbf{R}$ denotes the real numbers). One should probably take into account continuous cochains or something like that, but I would already be very happy to know if it is $0$ or not.

Of course this is equivalent to find a nontrivial crossed homomorphism $\mathbf{R}^\ast \to \mathbf{R}$, and I think that $\log$ or $\exp$ can help here, but I am stuck.

Any help is appreciated!