I just saw the continuity equation, in a manuscript, written as $$\frac{\partial \log \rho }{\partial t} + \vec v \cdot \nabla \log \rho= - \nabla \cdot \vec v.$$ Now, just calculating the derivatives of $\log$, and multiplying by $\rho$, this comes back to the familiar $$\frac{\partial \rho }{\partial t} + \nabla \cdot (\rho \vec v)= 0.$$ But I am curious: what would be the reason to write it in that log-form?
The log-form also appears on page 53 (pdf page 69) in this manual. And page 2 here explains about the same as tpg2114's answer.
Answer
Without seeing the manuscript in question, the most obvious reason why is when extremely large ranges of density are expected. If the density varies by orders of magnitude, say in an astrophysics setting, then the log form would keep the numbers in similar scale making it more numerically tractable.
Similar treatment is done for the partial density equations in chemically reacting flows.
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