  We introduce and study a log discrepancy function on the space of
semivaluations centered on an integral noetherian scheme of positive
characteristic. Our definition shares many properties with the analogue in
characteristic zero; we prove that if log resolutions exist, then our
definition agrees with previous approaches to log discrepancies of
semivaluations that these resolutions. We then apply this log discrepancy to a
variety of topics in singularity theory over fields of positive characteristic.
Strong F-regularity and sharp $F$-purity of Cartier subalgebras are detected
using positivity and non-negativity of log discrepancies of semivaluations,
just as Kawamata log terminal and log canonical singularities are defined using
divisorial log discrepancies, making precise a long-standing heuristic. We
prove, in positive characteristic, several theorems of Jonsson and Mustata in
characteristic zero regarding log canonical thresholds of graded sequences of
ideals. Along the way, we give a valuation-theoretic proof that asymptotic
multiplier ideals are coherent on strongly F-regular schemes.
