= Solution
The <ordinary least squares> equations for model 2 are
$$
X^T(\log T-X\widehat\alpha)=0.
$$
For the Gamma <generalized linear model>, $V(\mu)=\mu^2$ and $d\mu/d\eta=\mu$, so its score equation under the logarithmic link is
$$
X^T\left(\frac{T}{\mu}-\mathbf1\right)=0,
\qquad \mu_i=e^{x_i^T\widehat\gamma}.
$$
When $T_i$ is close to $\mu_i$,
$$
\frac{T_i}{\mu_i}-1
=e^{\log T_i-\log\mu_i}-1
\simeq\log T_i-x_i^T\widehat\gamma
$$
by the first-order <Taylor expansion> of the exponential function. The Gamma score equations then become the model-2 normal equations, so their coefficient estimates are close.
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