  We show that classical Chern classes from higher ($p$-adic) $K$-theory to
syntomic cohomology extend to logarithmic syntomic cohomology. These Chern
classes are compatible -- in a suitable sense -- with addition, products, and
$\lambda$-operations. They are also compatible with the canonical Gysin
sequences and, via period maps, with logarithmic \'etale Chern classes.
Moreover, they induce logarithmic crystalline Chern classes. This uses as a
critical new ingredient the recent comparison of syntomic cohomology with
$p$-adic nearby cycles and $p$-adic motivic cohomology.
