  Minima of the log-multiphase variational integral $$
  w \mapsto \int_{\Omega} \left[|Dw|\log(1+|Dw|) + a(x)|Dw|^q +
b(x)|Dw|^s\right] \, {\rm d}x\,, $$ have locally H\"older continuous gradient
under sharp quantitative bounds linking the growth powers $(q,s)$ to the
H\"older exponents of the modulating coefficients $a(\cdot)$ and $b(\cdot)$
respectively.
