The geodesic distance between points in SO(3) specified by the rotation matrices $R_1$ and $R_2$ is stated to be $d_g = \Vert\log\mathbf{R_1R_2}^\mathrm{T}\Vert_\mathrm{F}$, for example on Wikipedia.
However, I found empirically that this expression is missing a factor of ${1\over\sqrt{2}}$ in order for distances to be restricted to $[0,\pi]$ (intuitively, the maximum arc length on the unit sphere).
The same Wikipedia entry also notes that $\Vert\log\mathbf{R}\Vert_\mathrm{F} = {1\over\sqrt{2}}\vert{\theta}\vert$, which suggests the missing factor if one of the rotations is the identity matrix. However, no proof or explanation of the $\sqrt{2}$ is suggested.
How can I derive the factor?