Let $X$ be a complex normal projective variety, let $|L|$ be a non empty linear series on $X$ and let $b(|L|)$ be its base ideal. Suppose $f:X'\rightarrow X$ is a log resolution of the ideal $b(|L|)$.

Is $f$ a log resolution of the linear series $|L|$ (even if $X$ is not smooth)?

If it is do you have a proof or a reference for this?