The purpose of this paper is to prove in wide generality the \(C_{st}\) conjecture (due to Fontaine and Jannsen) on \(p\)-adic étale and log. crystalline cohomology. Let \(K\) be a complete discrete valuation field of characteristic \(0\) with perfect residue field \(k\) of characteristic \(p > 0\) and \({\mathcal O}_{K}\) be the ring of integers of \(K\). There is a ring \(B_{st}\) requiring heavy definitions defined by Fontaine depending on the choice of a prime element \(\pi\) of \(K\). The conjecture \(C_{st}\) states that there is a canonical \(B_{st}\)-linear isomorphism \[ B_{st}\otimes_{{\mathbb{Q}}_{p}}H^{m}_{et}(X_{\overline{K}},{\mathbb{Q}}_{p})\simeq B_{st}\otimes_{K_{0}}H_{log.-crys}(X) \] preserving the action of the absolute Galois group \(\text{Gal}(\overline{K}/K)\), associated linear endomorphisms \(\phi\) and \(N\) and the filtration induced after tensoring with \(B_{dR}\) over \(B_{st}\). Here, \(K_{0}\) is the field of fractions of the ring \(W\) of Witt-vectors in \(k\). To prove this conjecture, the author develops and uses facts on the syntomic complex and \(p\)-adic vanishing cycles.