y = Ce^(kt)

y = your total cells after a given period of time (t)
C = the starting number of cells
k = growth constant

The cells double every half a day, so that means that they quadruple every whole day, so the growth rate is 400% per day.

So

4 = (1)e^(k)(1) (since t = 1 day)

ln (4) = k

k = ln (4) (our constant is the natural log of four)

so

y = (1)e^((ln 4)(t))

So we are looking for 1 billion cells, so we must solve for (t).

1,000,000,000 = (1)e^((ln 4)(t))

1,000,000,000 = e^((ln 4)(t))

ln (1,000,000,000) = (ln 4)t

t = ln (1,000,000,000) / ln (4)

Your mouse is dead in 15 days (14.948676427 to be almost-exact).

Calculus is fun.