Exponential growth is represented by: Population (P) = Initial Population (i) × e^(rt), where e is the natural logarithm, r = rate (%), and t = time (hours), or P = i•e^rt
We can make initial (i) = x
So when that doubles, the P = 2x
Rate (r) = 3.5%, or .035
P = i•e^rt --> 2x = x•e^(.035•t)
Divide both sides by x, then:
2 = e^(.035•t)
Take the natural logarithm (ln) of both sides: ln (2) = ln [e^(.035•t)]
ln (2) = .693 and the ln [e^y] is the same as y•1, where our y = .035•t•1 = .035•t
So now we have .693 = .035•t
Divide both sides by .035, which gives: t = 19.8 hours
We can check that by multiplying 19.8×.035 = .693, take e^.693 = 1.9997, or 2, so since 2x = x•2, we know that we have the correct answer of the bacteria population doubling in 19.8 hours.
