In modern treatments of statistical mechanics, the natural base is conventionally used for the Gibbs and Boltzmann entropy without careful justification. While I am aware that the properties of the Shannon entropy are invariant to the choice of base of the logarithm, I suspect that physicists might have careful theoretical justifications for this particular choice. >Question: Might there be a sound theory behind this convention? So far a couple reasons occurred to me: 1. Stirling's log-factorial approximation: \begin{equation} \ln N! \sim N \cdot \ln N - N \tag{1} \end{equation} which is an essential tool in statistical mechanics. 2. The exponential function diagonalises the derivative operator: \begin{equation} \forall \lambda \in (0,1), \frac{\partial}{\partial H} \exp(\lambda \cdot H) = \lambda \cdot \exp(\lambda \cdot H) \tag{2} \end{equation} which may be useful whenever one wants to analyse variations in the exponential of entropy. The advantage of the exponential of entropy is that it is parameterisation invariant [as pointed out by Tom Leinster on a related question](https://mathoverflow.net/questions/398217/geometric-interpretations-of-the-exponential-of-entropy). As there might be subtle reasons I have ignored, any useful references on this question are more than welcome. ## References: 1. von Neumann, John (1932). Mathematische Grundlagen der Quantenmechanik (Mathematical foundations of quantum mechanics) Princeton University Press., . ISBN 978-0-691-02893-4. 2. Landau and Lifshitz. Statistical Physics. Butterworth-Heinemann. 1980. 3. David J.C. MacKay. Information Theory, Inference and Learning Algorithms. Cambridge University Press 2003.