  Consider $(M,g)$ a compact, boundaryless Riemannian manifold admitting an
Anosov geodesic flow. Let $\epsilon < \min\{1, \frac{\lambda_{\max}}{2}\}$
where $\lambda_{\max}$ is the maximal expansion rate for $(M,g)$. We study the
semiclassical measures $\mu_{sc}$ of $\epsilon$-logarithmic modes, which are
quasimodes spectrally supported in intervals of width $\epsilon
\frac{\hbar}{|\log \hbar|}$, of the Laplace-Beltrami operator on $M$. Under a
technical assumption on the support of a signed measure related to $\mu_{sc}$,
we show that the lower bound for the Kolmogorov-Sinai entropy of $\mu_{sc}$
generalizes that of Ananthamaran-Koch-Nonnenmacher. A feature of our results is
the quantitative nature of our bounds on hyperbolic manifolds.
  Via previous results of the author with Nonnenmacher and Silberman, we
provide another demonstration of the existence of $\epsilon$-logarithmic modes
whose semiclassical measures possess zero entropy ergodic components yet have
other components with positive entropy.
