  We model driven, compressible, isothermal, turbulence with Mach numbers
ranging from the subsonic ($\mathcal{M} \approx 0.65$) to the highly supersonic
regime ($\mathcal{M}\approx 16 $). The forcing scheme consists both solenoidal
(transverse) and compressive (longitudinal) modes in equal parts. We find a
relation $\sigma_{s}^2 = \mathrm{b}\log{(1+\mathrm{b}^2\mathcal{M}^2)}$ between
the Mach number and the standard deviation of the logarithmic density with
$\mathrm{b} = 0.457 \pm 0.007$. The density spectra follow
$\mathcal{D}(k,\,\mathcal{M}) \propto k^{\zeta(\mathcal{M})}$ with scaling
exponents depending on the Mach number. We find $\zeta(\mathcal{M}) = \alpha
\mathcal{M}^{\beta}$ with a coefficient $\alpha$ that varies slightly with
resolution, whereas $\beta$ changes systematically. We extrapolate to the limit
of infinite resolution and find $\alpha = -1.91 \pm 0.01,\, \beta =-0.30\pm
0.03$. The dependence of the scaling exponent on the Mach number implies a
fractal dimension $D=2+0.96 \mathcal{M}^{-0.30}$. We determine how the scaling
parameters depend on the wavenumber and find that the density spectra are
slightly curved. This curvature gets more pronounced with increasing Mach
number. We propose a physically motivated fitting formula $\mathcal{D}(k) =
\mathcal{D}_0 k^{\zeta k^{\eta}}$ by using simple scaling arguments. The fit
reproduces the spectral behaviour down to scales $k\approx 80$. The density
spectrum follows a single power-law $\eta = -0.005 \pm 0.01$ in the low Mach
number regime and the strongest curvature $\eta = -0.04 \pm 0.02$ for the
highest Mach number. These values of $\eta$ represent a lower limit, as the
curvature increases with resolution.
