= Solution
Assume an <ideal-gas atmosphere> of constant mean molecular mass and approximately constant gravity. If temperature decreases linearly with altitude, write $dT/dz=-\Gamma$. Combining the ideal-gas law with <hydrostatic equilibrium> gives
$$
\frac{d\log P}{dz}=-\frac{\mu m_Hg}{k_BT},
\qquad
\frac{d\log P}{d\log T}=\frac{\mu m_Hg}{k_B\Gamma}.
$$
The endpoint conditions determine the power law directly:
$$
\boxed{
T(P)=T_t\left(\frac{P}{P_t}\right)^a
=T_b\left(\frac{P}{P_b}\right)^a,
\qquad
a=\frac{\log(T_b/T_t)}{\log(P_b/P_t)}}.
$$
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