How many hours will it take a culture of bacteria to increase from 20 to 2,000 if the growth rate per hour is 85%?

Jul 27, 2016

Approximately $\textcolor{g r e e n}{7.5 \text{ hours}}$

Explanation:

We are trying to determine the value of $\textcolor{b l u e}{h}$ in the compound growth formula
$\textcolor{w h i t e}{\text{XXX}} 20 \times {\left(1 + 0.85\right)}^{\textcolor{b l u e}{h}} = 2000$
or, simplified:
$\textcolor{w h i t e}{\text{XXX}} {1.85}^{\textcolor{b l u e}{h}} = 100$

Taking the ${\log}_{1.85}$ of both sides:
$\textcolor{w h i t e}{\text{XXX}} {\log}_{1.85} 100$

With the aid of my trusty spreadsheet:
$\textcolor{w h i t e}{\text{XXX}} h \approx 7.48582192$