Conditions for a log-subharmonic function $f$ on $D\in\mathbb{R}^n$ are described by Mochizuki in A Class of Subharmonic Functions and Integral Inequalities (2004).

The conditions are phrased in terms of an inequality for the volume average $A_p$ of $|f|^p$ and the surface average $M$ of $|f|$, in the form $A_p\leq M^p$ for any closed ball in $D$.
When $D\in\mathbb{R}^n$ and $n\geq 2$ the condition $A_{1+2/n}\leq M^{1+2/n}$ is sufficient for $\log|f|$ to be subharmonic. If $n=2$ this condition is also necessary, if $n>2$ it is not.

Conditions for a log-subharmonic function $f$ on $D\in\mathbb{R}^n$ are described by Mochizuki in A Class of Subharmonic Functions and Integral Inequalities (2004).

The conditions are phrased in terms of an inequality for the volume average $A_p$ of $|f|^p$ and the surface average $M$ of $|f|$, in the form $A_p\leq M^p$ for any closed ball in $D$.
If $n\geq 2$ the condition $A_{1+2/n}\leq M^{1+2/n}$ is sufficient for $\log|f|$ to be subharmonic. If $n=2$ this condition is also necessary, if $n>2$ it is not.