  For each positive characteristic multiple zeta value (defined by Thakur), the
first and third authors constructed a $t$-module together with an algebraic
point such that a specified coordinate of the logarithmic vector of the
algebraic point is a rational multiple of that multiple zeta value. The
objective of this paper is to use the Taylor coefficients of Anderson-Thakur
series and $t$-motivic Carlitz multiple star polylogarithms to give explicit
formulae for all of the coordinates of this logarithmic vector.
