Question

The growth of a colony of bacteria is modeled by the function below, where t is the

is time in hours after the culture is begun and f(t) is the number of bacteria present.
After how many hours will there be 2000 bacteria in the colony? Round your answer
to the nearest hundredth (two decimal places).
f(t) = 500e^0.195t

Answer 1

namely, what is "t" when f(t) = 2000

`\begin{array}{llll} \textit{Logarithm Cancellation Rules} \\\\ \stackrel{ \textit{we'll use this one} }{log_a a^x = x}\qquad \qquad a^(log_a (x))=x \end{array} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{f(t)}{2000}=500e^(0.195t)\implies \cfrac{2000}{500}=e^(0.195t)\implies 4=e^(0.195t) \\\\\\ \log_e(4)=\log_e\left( e^(0.195t)\right)\implies \log_e(4)=0.195t\implies \ln(4)=0.195t \\\\\\ \cfrac{\ln(4)}{0.195}=t\implies 7.11\approx t`