= Solution
At $X_e=1/2$, the left side of the <Saha equation> is two. With its rounded coefficient, the <recombination temperature> therefore satisfies
$$
2=3\times10^{-16}\frac{e^y}{y^{3/2}},\qquad
y-\frac32\ln y=\ln\left(\frac23\times10^{16}\right)\simeq36.
$$
Neglecting the logarithmic prefactor gives $y\sim36$ and
$$
\boxed{T_{\rm rec}\sim13.6/36\ {
m eV}\sim0.4\ {
m eV}}.
$$
This is the requested rough scale. Retaining the prefactor and the unrounded logarithm gives $y\simeq42.0$ and $T_{\rm rec}\simeq0.32$ eV; the displayed $0.4$ eV is not a high-accuracy numerical root of the <Saha equation>.
The small <baryon-to-photon ratio> means that radiation and the entropy of the free charged particles strongly favor <ionization>. A low mean <photon> energy does not eliminate the energetic tail, and ionized particles have a large phase-space advantage. The exponential binding factor must become very large before neutral atoms dominate this dilute gas. Accordingly $\mathcal B/T$ is of order forty rather than order one, so \b[recombination occurs well below the <hydrogen> binding energy].
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