  A log generic hypersurface in $\mathbb{P}^n$ with respect to a birational
modification of $\mathbb{P}^n$ is by definition the image of a generic element
of a high power of an ample linear series on the modification. A log
very-generic hypersurface is defined similarly but restricting to line bundles
satisfying a non-resonance condition. Fixing a log resolution of a product
$f=f_1\ldots f_p$ of polynomials, we show that the monodromy conjecture,
relating the motivic zeta function with the complex monodromy, holds for the
tuple $(f_1,\ldots,f_p,g)$ and for the product $fg$, if $g$ is log generic. We
also show that the stronger version of the monodromy conjecture, relating the
motivic zeta function with the Bernstein-Sato ideal, holds for the tuple
$(f_1,\ldots,f_p,g)$ and for the product $fg$, if $g$ is log very-generic.
