What is the impact on the state equation of a Gibbs ensemble if we change the base to 2. For example,
$$ \begin{align} &Z=\sum_i{2^{-\beta (E_i+pdV)}} & \text{partition function} \\ &dE=TdS-pdV & \text{state equation} \end{align} $$
Does the change of basis changes the state equation?
Changing this is the equivalent of changing the temperature of your system. This can be seen just by recasting the new basis into the old basis. First note that $$\gamma^x=(e^{\log\gamma})^x=e^{x \log\gamma},$$ where I choose $\gamma > 1$. Hence your partition function becomes $$Z=\sum_i{e^{-\log\gamma~\beta (E_i+pdV)}} = \sum_i{e^{-\beta' (E_i+pdV)}}.$$ Where we see it is unchanged, if we change our temperature such that $\beta' = \beta\log\gamma $. That is $T'\rightarrow \frac{T}{\log\gamma}$.
Hence the equation of state would be $$dE=T'dS-pdV=\frac{T}{\log \gamma}dS - pdV$$
