= Solution
Take the logarithm to base ten and write pressure in units of $1\,\mathrm{bar}$, so its argument is dimensionless. Continuity of the <atmospheric pressure-temperature profile> gives
$$
C=\frac{1400-500}{\log_{10}(1)-\log_{10}(10^{-2})},\qquad \boxed{C=450\,\mathrm{K}\ \text{per pressure decade}}.
$$
Thus
$$
\boxed{T(P)=\begin{cases}
500\,\mathrm{K},&P\leq10^{-2}\,\mathrm{bar},\\
1400\,\mathrm{K}+450\,\mathrm{K}\log_{10}(P/1\,\mathrm{bar}),&10^{-2}\,\mathrm{bar}<P<1\,\mathrm{bar},\\
1400\,\mathrm{K},&P\geq1\,\mathrm{bar}.
\end{cases}}
$$
If the logarithm means $\ln$, the equivalent constant is $C=900/\ln100\simeq195.4\,\mathrm{K}$. A plot of <temperature> against logarithmic <pressure> is vertical in each isothermal region and straight between the two endpoints.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/iii/paper-315-profile.png]
{title=Atmospheric <pressure>-<temperature> profile}
The processes can be organized by the supplied pressure ranges, although the exact boundaries require reaction rates, irradiation and mixing information:
* At $P>1\,\mathrm{bar}$, the dense $1400\,\mathrm{K}$ gas can approach <thermochemical equilibrium> because collisions and reactions are relatively rapid. Deep <carbon monoxide> and <molecular nitrogen> can provide reservoirs for transported material.
* At $10^{-2}<P<1\,\mathrm{bar}$, the falling <temperature> slows chemical conversion. Vertical transport can produce a <chemical quench level> when the <chemical relaxation time> crosses the <eddy mixing time>. <Horizontal chemical quenching> is also possible if dayside and nightside conditions differ. Suitable species can condense and undergo <atmospheric condensate rainout> where a saturation curve is crossed.
* At $P<10^{-2}\,\mathrm{bar}$, slow thermal chemistry permits a <quenched atmospheric mixing ratio> to survive. <Atmospheric photochemistry> can dominate where stellar ultraviolet photons penetrate, often at still lower pressures; <atmospheric haze> may form from its products. Extremely high layers can also experience <atmospheric escape>.
\b[The profile identifies plausible chemical regimes, but does not fix their transition pressures by itself.] In particular, cloud formation depends on the species-specific condensation curve, and ultraviolet processing depends on shielding.
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