Suppose $M$ is an object in the abelian category of mixed Tate motives over $\mathbb{Q}$, and it is an extension of $\mathbb{Q}(0)$ by $\mathbb{Q}(1)$ \begin{equation} 0 \rightarrow \mathbb{Q}(1) \rightarrow M \rightarrow \mathbb{Q}(0) \rightarrow 0. \end{equation} Suppose the Hodge realisation of $M$ is the one associated to $\log u, u \in \mathbb{Q}^*$, i.e. $u$ is a nonzero rational number. Suppose $p$ is an unramified prime of $M$, and in the $p$-adic realisation of $M$, what is the matrix associated to the geometric Frobenius?
The matrix must be of the form \begin{pmatrix} 1, ~~~0 \\ *, 1/p \end{pmatrix} but is the unknown $*$ in the matrix just the $p$-adic value of $\log u$, i.e. the $p$-adic logarithm valued at $u$.
Remark, I do not understand the $p$-adic realisations of the mixed Tate motives very well, so the statement of this question might not be very rigorous. References about the $p$-adic realisations of mixed Tate motives are welcomed.
