KöMaL 2023 C1762

Is there a positive prime number $p$ for which $\log_{p-2}(4p-11)=m,$ where the parameter $m$ denotes one of the digits of $2023$ ?

Official text KöMaL. Quoted with source.

Solutions

No solution is on olymmath yet. Official and human solutions come first when they arrive. AI solutions come after them, with a label.