Equivalence of Boltzmann and Clausius entropy

[Bobhawke]

I can't figure out how to show that the Clausius entropy

$$\Delta S= \int_{T_1}^{T_2} \frac{dQ}{T}$$

is equivalent to the Boltzmann entropy

$$S=kln \Omega$$

at thermal equilibrium.

Does anyone know how to do this?

 

[flatmaster]

Might want to see how it works for monoatomic ideal gas first, then look for a generalization.