Your equation is implicit in the derivative. We can work around the problem by taking the derivative of the equation, which makes it explicit in the second derivative. If I don't misunderstand your syntax, here is an attempt:
eq = e[t]^(4/10) == (((237/100)/e'[t]^ (6/100)) -
16/10)/(10 (1 - Log[e'[t]]/(138/10))^(7/100))
deq = Simplify@D[eq, t]
de1 = e'[1] /.
Solve[eq /. {t -> 1} /. e[1] -> 1/1000, e'[1], Reals][[1]] // N
sol = NDSolveValue[{deq, e[1] == 1/1000, e'[1] == de1},
e, {t, 1, 18000}]
Plot[sol[t], {t, 1, 18000}]
