= Solution
Write $\mathscr P_c(k,\tau)=k^3|\Delta_c(k,\tau)|^2/(2\pi^2)$ for the <dimensionless power spectrum>. A radiation-era mode enters the horizon at $\tau_h\sim k^{-1}$. The early superhorizon law gives $\mathscr P_c(k,\tau_h)\sim A$: the $\tau_h^4$ and $k^4$ factors cancel.
For $k\tau_{\rm eq}\ll1$, the mode remains outside the horizon throughout the radiation era. Its amplitude grows as $\tau^2$ both before and after equality, and continues with the same scale-independent growing factor even if it later enters during <matter domination>. Hence the low-wavenumber shape is preserved:
$$
\boxed{\mathscr P_c(k,\tau)\sim A\tau^4k^4,\qquad k\tau_{\rm eq}\ll1.}
$$
For $k\tau_{\rm eq}\gg1$, horizon entry occurs during <radiation domination>. Its subsequent logarithmic amplitude growth supplies a factor $\ln(\tau_{\rm eq}/\tau_h)\sim\ln(k\tau_{\rm eq})$ by equality. During the matter era the amplitude grows by $(\tau/\tau_{\rm eq})^2$. Squaring gives the <logarithmic high-wavenumber density-spectrum transfer>:
$$
\boxed{\mathscr P_c(k,\tau)\sim A\left(\frac{\tau}{\tau_{\rm eq}}\right)^4[\ln(k\tau_{\rm eq})]^2,\qquad k\tau_{\rm eq}\gg1.}
$$
The horizon-entry and equality matching fixes order-one coefficients and additive constants inside the logarithm. The displayed branches capture the requested leading spectrum and share its schematic normalization; their asymptotic forms must not be joined literally at $k\tau_{\rm eq}=1$, where the high-$k$ logarithm alone vanishes.
Thus the present <dimensionless power spectrum> rises as $k^4$ on large scales and only as $\ln^2 k$ on small scales, with a smooth bend near the <matter-radiation equality scale>, $k_{\rm eq}\sim\tau_{\rm eq}^{-1}$. The ordinary dimensional <cosmological density power spectrum> instead satisfies
$$
P_c(k)=\frac{2\pi^2}{k^3}\mathscr P_c(k)\propto\begin{cases}k,&k\ll k_{\rm eq},\\ k^{-3}\ln^2(k/k_{\rm eq}),&k\gg k_{\rm eq}.
\end{cases}
$$
It has a turnover near equality; the dimensionless spectrum does not. The following comparison of <dimensional and dimensionless matter spectra across equality> uses a smooth guide with the correct asymptotes, not an exact solution through the transition. Overall vertical normalization and the later scale-independent growth factor are suppressed.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/iii/paper-53-matter-spectrum.png]
{title=Present matter spectrum across equality: dimensionless power bends from k to the fourth power to logarithmic growth, while dimensional power turns over}
{height=440}
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