= Solution
At one loop, plot each inverse squared coupling $g_a^{-2}$ against $\log\mu$. The three Standard Model lines have slopes that do not pass through one common point. Above the superpartner threshold, additional scalar and fermion vacuum-polarization diagrams change the slopes: for example, an $SU(2)$ gauge-boson two-point function receives a loop from a left-handed Standard Model fermion doublet and an additional loop from its scalar superpartner, while the $SU(2)$ gauge multiplet adds a <gaugino> loop alongside the gauge-boson and ghost loops. With the <Minimal supersymmetric Standard Model> field content, the three resulting straight lines meet to good accuracy. This is <supersymmetric gauge coupling unification>.
The differential equation
$$
\frac{dg_a}{d\log\mu}=\beta_ag_a^3
$$
implies
$$
\frac{d}{d\log\mu}g_a^{-2}=-2\beta_a.
$$
Hence
$$
\boxed{g_a^{-2}(\mu)=g_a^{-2}(\mu_0)
-2\beta_a\log\frac\mu{\mu_0}},
$$
or
$$
\boxed{g_a(\mu)=\frac{g_a(\mu_0)}
{\sqrt{1-2\beta_ag_a^2(\mu_0)\log(\mu/\mu_0)}}}.
$$
Let $L=\log(M_{GUT}/M_Z)$. Equality of $g_1$ and $g_2$ at the unification scale gives
$$
g_1^{-2}(M_Z)-2\beta_1L
=g_2^{-2}(M_Z)-2\beta_2L,
$$
so
$$
\boxed{M_{GUT}=M_Z\exp\!\left[
\frac{g_1^{-2}(M_Z)-g_2^{-2}(M_Z)}{2(\beta_1-\beta_2)}
\right]}.
$$
Similarly,
$$
g_3^{-2}(M_Z)-g_2^{-2}(M_Z)
=2(\beta_3-\beta_2)L,
$$
while
$$
g_2^{-2}(M_Z)-g_1^{-2}(M_Z)
=2(\beta_2-\beta_1)L.
$$
Eliminating $L$ yields
$$
\boxed{g_3^{-2}(M_Z)=g_2^{-2}(M_Z)
+A\left[g_2^{-2}(M_Z)-g_1^{-2}(M_Z)\right]},
$$
with
$$
\boxed{A=\frac{\beta_3-\beta_2}{\beta_2-\beta_1}}.
$$
For the GUT-normalized MSSM one-loop coefficients proportional to $(33/5,1,-3)$, this is $\boxed{A=5/7}$.
Current coupling measurements approximately satisfy this MSSM relation and point to $M_{GUT}$ of order $10^{16}$ GeV, much more accurately than nonsupersymmetric one-loop running. Exact equality is not expected because two-loop evolution and threshold corrections from split superpartner and GUT-scale masses shift the lines; the absence so far of directly observed superpartners also prevents the threshold spectrum from being fixed experimentally.
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