# How long does it take the culture to double its mass if a bacterial culture which is growing exponentially increases from 5 g to 7 g in 11 hours?

Feb 22, 2015

This is an example of a exponential function.

The form is always $N = B \cdot {g}^{t}$
Where
$N$=new situation
$B$=begin
$g$=growth factor
$t$=time units

Now we fill in what we know:
$N = B \cdot {g}^{t} \to 7 = 5 \cdot {g}^{11} \to {g}^{11} = 7 / 5 = 1.4$
Now grow factor $g$ is the 11th root of $1.4$

You can do this on your calculator by $\text{1.4^(1/11)}$ or
${1.4}^{\frac{1}{11}} = 1.03106 \ldots$

The growth factor can then be used in the equation:
$N = B \cdot {g}^{t} \to 10 = 5 \cdot 1.03106 {\ldots}^{t} \to 1.03106 {\ldots}^{t} = 2$

From here on there are two ways to go:

(1) put this equation in your GC as Y1= and intersect with Y2=2

(2) Use logarithms:

$1.03106 {\ldots}^{t} = 2 \to \log 1.03106 {\ldots}^{t} = \log 2 \to$
$t \cdot \log 1.03106 \ldots = \log 2 \to t = \log \frac{2}{\log} 1.03106 \approx 22.3$

Answer : a bit more than 22 hours. This is the doubling time from any start situation.