While testing a new vaccine, a lab technician noticed that the number of bacteria reduced by half every minute. How long does it take for 3,000 bacteria to reduce to 375? It takes minutes for the number of bacteria to reduce from 3,000 to 375.

This is a geometric progression:

Tn=a₁r^ (n-1)

In this case:

Tn=a₁r^n

Because:

T₁ corresponds to 0 minutes ⇒T₁=3000 (1/2) ⁰=3000

T₂ corresponds to 1 minute ⇒ T²=3000 (1/2) ¹=1500 ...

Tn=375

a₁=3000

r=1/2

n=number of minutes

Therefore:

375=3000 * (1/2) ^n

375/3000 = (1/2) ^n

0.125 = (1/2) ^n

Ln 0.125=ln [ (1/2) ^n]

Ln 0.125=n Ln (1/2)

Ln 0.125=n [Ln1-Ln2]

Ln 0.125=n[0-Ln 2]

n=Ln 0.125 / (-Ln2)

n=3

answer: 3 minutes.