= Solution
Assume matter-dominated growth over the two epochs, so $D(z)\propto(1+z)^{-1}$. A fixed definition of the characteristic mass selects fixed peak height $\nu_*$. Therefore $D(z)\sigma(M_*)$ is constant. With $\sigma(M)\propto M^{-s}$, the observed ratios give
$$
16=\frac{M_*(3)}{M_*(7)}=\left(\frac{D(3)}{D(7)}\right)^{1/s}=2^{1/s},\qquad s=\frac14.
$$
Since a <spherical top-hat window function> has $M\propto R^3$, the rms slope is
$$
\boxed{m=-3s=-\frac34.}
$$
For a <cosmological density power spectrum> $P(k)\propto k^n$, the <smoothed matter density variance> is
$$
\sigma^2(R)=\frac1{2\pi^2}\int_0^\infty k^2P(k)W(kR)^2\,dk\propto R^{-(n+3)},\qquad W(q)=\frac{3(\sin q-q\cos q)}{q^3}.
$$
The dimensionless integral converges for $-3<n<1$. Thus $m=-(n+3)/2$, and
$$
\boxed{n=-2m-3=-\frac32.}
$$
For reference, substituting $s=1/4$ into the differential <Press-Schechter halo mass function> gives a low-mass prefactor $M^{-7/4}$ and an exponential argument proportional to $M^{1/2}$, so its displayed-form exponents are $\alpha=-7/4$, $\beta=1/2$. The prefactor normalization varies with epoch.
The effective galactic-scale index $-3/2$ need not equal the primordial nearly scale-invariant index one. The <cosmological transfer function> modifies perturbations between the primordial epoch and galaxy formation: modes entering the horizon in the radiation-dominated era grow differently from larger modes. Schematically $P_m(k,z)\propto D(z)^2k^{n_s}T(k)^2$, with $n_s\simeq1$, while the <cold-dark-matter transfer function> decreases toward small scales. The result is a much redder late-time <cosmological density power spectrum>, with a scale-dependent slope. \b[The inferred $-3/2$ is a local power-law approximation to the processed matter spectrum, not a measurement of the primordial index.]
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