  In this paper we prove that if $\{\varphi_i(x)=\lambda x+t_i\}$ is an
equicontractive iterated function system and $b$ is a positive integer
satisfying $\frac{\log b}{\log |\lambda|}\notin\mathbb{Q},$ then almost every
$x$ is normal in base $b$ for any non-atomic self-similar measure of
$\{\varphi_i\}$.
