I have the following equation;
$$TS-PV =\alpha\cdot N\cdot K_B\cdot \left(\frac NV\right)^\frac23\cdot e^{\frac{2S}{3N\cdot K_b}}$$
which i have to reduce to the Ideal gas Law;
$$PV = NK_bT$$ I have been given the equation derived from Maxwell thermodynamic relations. The issue is that the exponential must be reduced and the only method is to get the $\ln()$ across the entire equation. I then have $\ln()$ and to cancel I need to get the exponential across the equation, but then I will have the exponential again and the cycle continues!
Regarding the entropy, S, they should cancel on both sides of the equation.