As Matt F. points out, we could just absorb a change of basis of the logarithm into the coefficient. The reason that is not convenient in physics is that we would like the same coefficient $k$ to appear in the ideal gas law $pV=NkT$.

Note that in physics entropy changes can be defined in thermodynamic rather than statistical terms, as the absorbed heat in a reversible transition ($\Delta S=\int dQ/T$). So any change in the base of the logarithm has to be compensated by a change in the coefficient $k\mapsto k'$. And if we would define the entropy $S=k'\log_2 W$ using the base-2 logarithm, the ideal gas law would read $pV\ln 2=Nk'T$.

As Matt F. points out, we could just absorb a change of base of the logarithm into the coefficient. The reason that is not convenient in physics is that we would like the same coefficient $k$ to appear in the ideal gas law $pV=NkT$.

Note that in physics entropy changes can be defined in thermodynamic rather than statistical terms, as the absorbed heat in a reversible transition ($\Delta S=\int dQ/T$). So any change in the base of the logarithm has to be compensated by a change in the coefficient $k\mapsto k'$. And if we would define the entropy $S=k'\log_2 W$ using the base-2 logarithm, the ideal gas law would read $pV\ln 2=Nk'T$.

Q: Why does the Boltzmann entropy $S=k\ln W$ use the natural logarithm?

A: To ensure that the same coefficient $k$ appears in the ideal gas law $pV=NkT$.

Note that in physics entropy changes can be defined in thermodynamic rather than statistical terms, as the absorbed heat in a reversible transition ($\Delta S=\int dQ/T$). So any changes in the base of the logarithm have to be compensated by a change in the coefficient $k\mapsto k'$. And if we would define the entropy $S=k'\log_2 W$ using the base-2 logarithm, the ideal gas law would read $pV\ln 2=Nk'T$.

As Matt F. points out, we could just absorb a change of basis of the logarithm into the coefficient. The reason that is not convenient in physics is that we would like the same coefficient $k$ to appear in the ideal gas law $pV=NkT$.

Note that in physics entropy changes can be defined in thermodynamic rather than statistical terms, as the absorbed heat in a reversible transition ($\Delta S=\int dQ/T$). So any change in the base of the logarithm has to be compensated by a change in the coefficient $k\mapsto k'$. And if we would define the entropy $S=k'\log_2 W$ using the base-2 logarithm, the ideal gas law would read $pV\ln 2=Nk'T$.