Question 184656
the number of bacteria in a culture increased sixfold in 10 hours. Assuming natural growth, how long did it take their number to double?
<pre><font size = 4 color = "indigo"><b>
Your textbook and teacher may be using different letters from the
ones I use, if so you can change the letters to yours.

{{{P = P[0]e^(rt)}}}

{{{P}}} has gone from {{{P[0]}}} to {{{6*P[0]}}} while t increased to {{{10}}}.  
So we substitute {{{6*P[0]}}} for {{{P}}} and {{{10}}} for {{{t}}}:

{{{6*P[0] = P[0]e^(r*10)}}}

Divide both sides by {{{P[0]}}}

{{{6 = e^(10r)}}}

Change to natural log form:

{{{10r = ln(6)}}}

{{{r = ln(6)/10}}}

{{{r = .1791759469}}}

So we take the original formula
and substitute {{{.1791759469}}}
for r.

{{{P = P[0]e^(rt)}}}
{{{P = P[0]e^(.1791759469t)}}}

Now we want to find what {{{t}}} is when {{{P=2*P[0]}}}

---
{{{2*P[0] = P[0]e^(.1791759469t)}}}

{{{2*P[0] = P[0]e^(.1791759469t)}}}

Divide both sides by {{{P[0]}}}

{{{2 = e^(.1791759469t)}}}

Change to natural log form:

{{{.1791759469t = ln(2)}}}

{{{t = ln(2)/.1791759469}}}

{{{t = 3.868528072hours}}}

Edwin</pre>