= Solution
The same gas-pressure <stellar homology> estimates give $T\propto\mu M/R_*$ and $\rho\propto M/R_*^3$. With <Kramers' opacity law> $\kappa\propto\rho T^{-7/2}$, <radiative diffusion> gives
$$
L\propto\frac{R_*T^4}{\kappa\rho}
\propto\frac{R_*T^{15/2}}{\rho^2}
\propto\mu^{15/2}M^{11/2}R_*^{-1/2}.
$$
Consequently the claimed exponent five additionally requires the approximate low-mass <main-sequence mass-radius relation> $R_*\propto M$ and fixed composition:
$$
\boxed{R_*\propto M\quad\Longrightarrow\quad L\propto M^5.}
$$
The supplied <opacity> and pressure assumptions alone do not establish that <mass-radius relation>. For example, a homologous fixed-radius family obeys those scaling assumptions but gives $L\propto M^{11/2}$, not $M^5$. More generally $R_*\propto M^q$ gives $L\propto M^{(11-q)/2}$. Very low-mass <stars> can be convective, where this radiative model ceases to apply.
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