  We prove that the Fourier transform of a self conformal measure on
$\mathbb{R}$ decays to $0$ at infinity at a logarithmic rate, unless the
following holds: The underlying IFS is smoothly conjugated to an IFS that both
acts linearly on its attractor and contracts by scales that are not
Diophantine. Our key technical result is an effective version of a local limit
Theorem for cocycles with moderate deviations due to Benoist-Quint (2016), that
is of independent interest.
