  We introduce formulas for the logarithms of Drinfeld modules using a
framework recently developed by the second author. We write the logarithm
function as the evaluation under a motivic map of a product of rigid analytic
trivializations of $t$-motives. We then specialize our formulas to express
special values of Goss $L$-functions as Drinfeld periods multiplied by rigid
analytic trivializations evaluated under this motivic map. We view these
formulas as characteristic-$p$ analogues of integral representations of
Hasse-Weil type zeta functions. We also apply this machinery for Drinfeld
modules tensored with the tensor powers of the Carlitz module, which serves as
the Tate twist of a Drinfeld module.
