  We prove a logarithmic base change theorem for pushforwards of
pluri-canonical bundles and use it to deduce that positivity properties of log
canonical divisors descend via smooth projective morphisms. As an application,
for a surjective morphism $f:X\to Y$ with $\kappa(X)\ge 0$ and $-K_Y$ big, we
prove $Y\setminus \Delta(f)$ is of log general type, where $\Delta(f)$ is the
discriminant locus. In particular, when $Y=\mathbb{P}^n$ we have $\dim
\Delta(f)=n-1$ and $\mathrm{deg}\,\Delta(f)\ge n+2$, generalizing the case
$n=1$ proved by Viehweg-Zuo. In addition, we prove Popa's conjecture on the
superadditivity of the logarithmic Kodaira dimension of smooth algebraic fiber
spaces over bases of dimension at most three and analyze related problems.
