= Solution
Use the classical <Faber-Jackson relation> $L\propto\sigma^4$ as the explicitly requested extrapolation. The supplied host fixes its normalization:
$$
L_d=10^{11}\left(\frac{10}{300}\right)^4L_\odot=1.23\times10^5L_\odot.
$$
With the mass derived using circular-orbit <Jacobi truncation of two singular isothermal spheres>, the dynamical <mass-to-light ratio> is
$$
\Upsilon_d=\frac{5.05\times10^7}{1.23\times10^5}\frac{M_\odot}{L_\odot}=4.09\times10^2\frac{M_\odot}{L_\odot}.
$$
Hence to one significant figure
$$
\boxed{L_d\simeq10^5L_\odot,\qquad \Upsilon_d\simeq4\times10^2M_\odot/L_\odot.}
$$
The rough equal-density truncation convention gives $\Upsilon_d\simeq6\times10^2M_\odot/L_\odot$ instead. Both conclusions indicate a very large dynamical <mass-to-light ratio> compared with a normal old stellar population.
Within their bright stellar bodies, typical giant <elliptical galaxies> have optical dynamical <mass-to-light ratios> of order a few to ten in solar units, with variations in band, <initial mass function>, and <dark matter>. Large outer apertures can give substantially larger ratios. Many <dwarf spheroidal galaxies> have much larger inferred ratios, from tens to hundreds and, for some very faint systems, still larger values. Thus the computed ratio is plausible as the scale of a very dark-matter-dominated faint dwarf, not as a purely stellar ratio. Dynamical inference depends on member identification, binary motions, equilibrium, and the assumed mass profile.
The stipulated <Faber-Jackson relation> is not a reliable universal extrapolation down to every <dwarf spheroidal galaxy>. Also, the <fundamental plane of elliptical galaxies> contains radius and <surface brightness> as well as <velocity dispersion>; the two-variable <Faber-Jackson relation> is a projection of that information, not an interchangeable definition of the full plane.
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