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Stellar evolution temperature gradient - why the logarithm?

This expression comes from considering what happens to a volume of gas moving a small distance $dr$. Such a movement happens in the "dynamical timescale", which is much, much smaller (~hours)...
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Virial theorem, contraction, and stars getting hotter

The gravitational energy gets more negative, so the right hand side of your equation gets numerically larger. Since the left hand side is (roughly) proportional to temperature, then the temperature ...
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