If I understood well, You just have to drag the layer in which particular is applied do 5 seconds, so the explosion only starts at 5 seconds.
But if you applied the particular effect at the same layer, create a keyframe for the number of particles to be generated to “0” at 5 seconds and at the next keyframe change this number to “200” or whaterver you want it to be.
For logo explosions you should also condider the “Shatter” effect and choose the shape to “Glass” and increase the number of “repetitions”, it sumulates good explosions but no smoke.
I hope it helps.