= Solution
Using the requested exponential law with a temperature-independent prefactor,
$$
\ln\frac{\sigma(420)}{\sigma(400)}
=\frac{E_g}{2k_B}\left(\frac1{400}-\frac1{420}\right).
$$
The conductivity ratio is two, so
$$
\boxed{E_g=\frac{2k_B\ln2}{1/400-1/420}}
=\frac{2(8.61733\times10^{-5})\ln2}{1/400-1/420}\,\mathrm{eV},
$$
$$
\boxed{E_g\simeq1.0035\,\mathrm{eV}\simeq1.00\,\mathrm{eV}}.
$$
The denominator is positive, as required for a positive gap and rising conductivity. If one instead retains the constant-DOS factor $T$ while keeping mobility constant, the inferred gap becomes
$$
E_g=\frac{2k_B\ln[2(400/420)]}{1/400-1/420}
\simeq0.933\,\mathrm{eV}.
$$
That is a different prefactor model; \b[$1.00\,\mathrm{eV}$ is the estimate from the relation specified for this part.]
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