= Solution
Write $t=\log\mu$ and use $\alpha_i=g_i^2/(4\pi)$. The <chain rule> turns the <renormalization-group beta function> into
$$
\frac{d\alpha_i}{dt}=\frac{g_i}{2\pi}\frac{dg_i}{dt}=\frac{\beta_i}{2\pi}\alpha_i^2,
\qquad \frac{d\alpha_i^{-1}}{dt}=-\frac{\beta_i}{2\pi}.
$$
Integrating gives the <one-loop inverse gauge coupling evolution>
$$
\boxed{\alpha_i^{-1}(\mu)=\alpha_i^{-1}(M_Z)-\frac{\beta_i}{2\pi}\log\frac\mu{M_Z}.}
$$
This expression applies over a range with unchanged active field content and in the perturbative regime. It uses $\log\mu$, not $\log\mu^2$.
For <gauge coupling unification>, define the normalized inverse couplings $A_1=3/(5\alpha_1)$, $A_2=1/\alpha_2$, $A_3=1/\alpha_3$, and slopes $b_1=(3/5)\beta_1$, $b_2=\beta_2$, $b_3=\beta_3$. At the common scale each inverse coupling is $A_U$. With $L=\log(M_{\mathrm{GUT}}/M_Z)$,
$$
A_i(M_Z)=A_U+\frac{b_i}{2\pi}L.
$$
Subtract the equations for factors 1 and 2 to solve $L=2\pi(A_1-A_2)/(b_1-b_2)$, then insert this into the difference of factors 3 and 2:
$$
\boxed{\frac1{\alpha_3(M_Z)}=\frac1{\alpha_2(M_Z)}+
\frac{\beta_3-\beta_2}{(3/5)\beta_1-\beta_2}
\left[\frac3{5\alpha_1(M_Z)}-\frac1{\alpha_2(M_Z)}\right].}
$$
This <one-loop normalized hypercharge unification> relation requires $b_1\ne b_2$. A scale above $M_Z$ also requires the inferred $L$ to be positive. If those two slopes coincide, unification instead requires $A_1=A_2$ and their difference cannot fix the scale.
In the convention $Q=T_3+Y$, one matter generation and one <Higgs doublet> have the following <Standard Model representations>. <Spin> is in units of $\hbar$; L/R describe <chirality>, not the <spin> projection. The $U(1)_Y$ entry is its <hypercharge> eigenvalue, defining the one-dimensional <representation> $e^{iY\omega}$.
|| Field
|| <Spin>
|| <Chirality>
|| $Y$
|| $SU(2)_L$
|| $SU(3)_c$
| $Q_L=(u_L,d_L)^T$
| $1/2$
| Left
| $1/6$
| $\mathbf2$
| $\mathbf3$
| $u_R$
| $1/2$
| Right
| $2/3$
| $\mathbf1$
| $\mathbf3$
| $d_R$
| $1/2$
| Right
| $-1/3$
| $\mathbf1$
| $\mathbf3$
| $L_L=(\nu_{eL},e_L)^T$
| $1/2$
| Left
| $-1/2$
| $\mathbf2$
| $\mathbf1$
| $e_R$
| $1/2$
| Right
| $-1$
| $\mathbf1$
| $\mathbf1$
| $H=(H^+,H^0)^T$
| $0$
| Not applicable
| $1/2$
| $\mathbf2$
| $\mathbf1$
There is no <right-handed neutrino> in the minimal field content being counted. Right-handed matter can alternatively be written as left-handed charge-conjugate <Weyl fermions>, in conjugate colour <representations> with opposite <hypercharges>. The quadratic indices and squared charges in the beta coefficients are unchanged.
For $SU(3)$, one generation supplies two colour fundamentals from $Q_L$ and one each from $u_R,d_R$. With three generations there are $n_f=3(2+1+1)=12$ <Weyl fermions> counted as fundamental copies, and no coloured scalar. Thus
$$
\beta_3=-\frac{11}{3}\,3+\frac13\,12=\boxed{-7}.
$$
For $SU(2)$, the three colours of $Q_L$ give three weak-doublet copies and $L_L$ gives one. There are $n_f=3(3+1)=12$ doublets across three generations and $n_s=1$ complex <Higgs doublet>, giving
$$
\beta_2=-\frac{11}{3}\,2+\frac13\,12+\frac16=\boxed{-\frac{19}{6}}.
$$
These are <spectator multiplicities in gauge beta functions>: the colour index multiplies the weak contribution, while the weak index multiplies the colour contribution. Each two-component fermion is counted once; it must not be counted again as a separate <antiparticle>.
For $U(1)_Y$, sum squared <hypercharges> over all colour and weak components. One generation gives
$$
\sum_{f,\,\text{one generation}}Y_f^2
=6\left(\frac16\right)^2+3\left(\frac23\right)^2+3\left(-\frac13\right)^2+2\left(-\frac12\right)^2+(-1)^2=\frac{10}{3}.
$$
The two complex Higgs components give $\sum_sY_s^2=2(1/2)^2=1/2$. There is no Abelian gauge self-interaction contribution. Therefore
$$
\beta_1=\frac23\,3\,\frac{10}{3}+\frac13\,\frac12=\boxed{\frac{41}{6}},\qquad
\boxed{(\beta_1,\beta_2,\beta_3)=\left(\frac{41}{6},-\frac{19}{6},-7\right).}
$$
The Higgs is counted only once for the whole model, whereas the fermionic generation is repeated three times. These are the <Standard Model one-loop gauge coefficients> in the unnormalized <hypercharge> convention. The grand-unified normalization instead has $b_1=41/10$. The ratio of slopes in the unification relation is $(\beta_3-\beta_2)/(b_1-\beta_2)=-115/218$.
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