  We use the Lewis and Riesenfeld invariant method [\textit{J. Math. Phys.}
\textbf{10}, 1458 (1969)] and a unitary transformation to obtain the exact
Schr\"{o}dinger wave functions for time-dependent harmonic oscillators
exhibiting log-periodic-type behavior. For each oscillator we calculate the
quantum fluctuations in the coordinate and momentum as well as the quantum
correlations between the coordinate and momentum. We observe that the
oscillator with $m=m_0t/t_0$ and $\omega= \omega_0t_0/t$, which exhibits an
exact log-periodic oscillation, behaves as the harmonic oscillator with $m$ and
$\omega$ constant.
