[?] Solve and Plot numerically this differential equation?

Questions
1

Consider the following code:

e1 = 0.001;
e^0.4 = {{2.37/de^0.06} - 1.6}/10 {1 - Log[de]/13.8}^0.7

the initial value is e1=0.001 for t0=1 and the time period is between {t,1,18000}

2

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}]
3

Thank you very much for your aswer!!!it works perfect!!!