Natural logarithm convention

A common convention in thermodynamics and statistical mechanics is that $\log$ means the natural logarithm:

unless another base is explicitly stated.

Where it appears

This convention is built into standard formulas such as:

• Boltzmann entropy: $S = k_B \ln \Omega,$ where $\Omega$ is a count (or suitably normalized measure) of accessible microstates.
• Helmholtz free energy (canonical ensemble): $F = -k_B T \ln Z,$ where $Z$ is the partition function.
• Free energy density in the thermodynamic limit: $f = -\frac{1}{\beta}\lim \frac{1}{|\Lambda|}\ln Z_\Lambda, \quad \beta = \frac{1}{k_B T}.$ (See thermodynamic limit convention .)

Why natural logs are the default

Changing the logarithm base rescales results by a constant factor:

If one insisted on base $b$, formulas like $S=k_B\log_b\Omega$ would amount to replacing $k_B$ by $k_B/\ln b$. Using $\ln$ keeps the usual physical value and units conventions for $k_B$ without extra conversion factors.

(When comparing with information theory, note that entropies may be measured in “bits” using base $2$; the conversion is a constant factor.)