Discrete Logarithm Notation
$\log_n(x)$ means the base-$n$ logarithm of $x$. It returns a value $y$ such that $n ^ y = x$.

Since you're working with discrete logarithms, the base will probably be the generator $g$ for your group.

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$L_{13}(x)$ appears to be a shorthand way of writing the above. The only way to find out otherwise would be to consult the notation definitions from the paper (or author) if possible, because it could mean something else entirely or something more specific.