Q = Q0e−5t
Sloppy equation ! What is the "0" between Q and e doing there ?
1 - Half-life: 1/2Q = Q * e^(-5t), solve for t
1/2 = 1 * e^(-5t)
1/2 = e^(-5t)
Take the natural log of both sides
t = ln(2) / 5 ≈0.138629 - this is the half-life of Q in seconds, or minutes or hours, days, years...etc.
2 - Doubling Period: This does NOT make sense the way you have your equation written, because it will give you the value of t = 0