= Solution
Use the canonically normalized Abelian coupling $g_1=-\sqrt{5/3}\,g'$, with $g_2=g$ and $g_3=g_s$, and write $\alpha_i=g_i^2/(4\pi)$. The sign in $g_1$ is a generator convention; its square is what enters running and matching. Let $\alpha_5=g_5^2/(4\pi)$ and $L_X=\log(M_X^2/\mu^2)$. To leading order,
$$
\frac{d}{d\log\mu^2}\frac1{\alpha_i}=\frac{b_i}{4\pi},\qquad\frac1{\alpha_i(\mu)}=\frac1{\alpha_5}-\frac{b_i}{4\pi}L_X.
$$
Use exactly the common-flavor coefficients stipulated for this model:
$$
b_3=11-\frac{2n_f}{3},\qquad b_2=\frac{22}{3}-\frac{2n_f}{3},\qquad b_1=-\frac{2n_f}{3}.
$$
The same $n_f$ cancels from their differences. In particular $b_3-b_2=11/3$ and $b_2-b_1=22/3$, so eliminating the unified coupling gives
$$
\frac1{\alpha_2}-\frac1{\alpha_s}=\frac{11}{12\pi}L_X,\qquad\frac1{\alpha_1}-\frac1{\alpha_2}=\frac{22}{12\pi}L_X=2\left(\frac1{\alpha_2}-\frac1{\alpha_s}\right).
$$
At the low reference scale, $e=g\sin\theta_W=g'\cos\theta_W$. With $w=\sin^2\theta_W$ this means
$$
\alpha_2^{-1}=\frac w\alpha,\qquad\alpha_1^{-1}=\frac{3(1-w)}{5\alpha}.
$$
The last difference equation becomes $3(1-w)/(5\alpha)-w/\alpha=2(w/\alpha-1/\alpha_s)$. Solving gives
$$
\boxed{\sin^2\theta_W=\frac16+\frac{5\alpha}{9\alpha_s}.}
$$
Substitute this into the first difference equation to obtain
$$
\boxed{\log\frac{M_X}{\mu}=\frac\pi{11}\left(\frac1\alpha-\frac{8}{3\alpha_s}\right),\qquad M_X=\mu\exp\!\left[\frac\pi{11}\left(\frac1\alpha-\frac{8}{3\alpha_s}\right)\right].}
$$
All low-scale couplings on the right must be evaluated at the same specified $\mu$. A dimensionful $M_X$ cannot be fixed by dimensionless coupling values without such a reference scale. The result is independent of the common $n_f$ in these differences, although $\alpha_5$ is not. An inferred $M_X>\mu$ requires $\alpha^{-1}>8/(3\alpha_s)$, and perturbative matching requires a positive small $\alpha_5$.
This is the <one-loop SU(5) prediction with common flavor coefficients>. It is the prediction of the supplied simplified coefficient model, not the full matter-and-Higgs calculation for a realistic <SU(5) grand unified theory>. Actual <representations>, intermediate thresholds and higher-loop terms change the running and require the corresponding matching calculation.
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