# Population growth logarithm

• Jan 14th 2009, 05:06 AM
mattty
Population growth logarithm
The question is: The bacterial population in a certain culture is given by $\displaystyle P=I(1.6)^t$ where $\displaystyle t$ is in hours and $\displaystyle I$ is the initial population (when $\displaystyle t=0$). Determine how long it takes for this population to double in size.

So, from this, i've worked out that $\displaystyle P=2I$ and then i took a stab at rearranging the equation into the form $\displaystyle x={log_by}$ and got
$\displaystyle t=I\log_{1.6}2I$

Now, although I believe this to be the right answer something isn't ringing true since natural logarithms didn't come into it at all. I've always been told that population growth was one of the main things natural logs were for outside of calculus.

Any help or suggestions would be greatly appreciated.
• Jan 14th 2009, 05:33 AM
galactus
Don't over complicate it.

Since it doubles, set P=2I. That way, the I's cancel and you have:

$\displaystyle 2=(\frac{8}{5})^{t}$

Now, take log of both sides:

$\displaystyle ln(2)=t\cdot ln(\frac{8}{5})$

$\displaystyle t=\frac{ln(2)}{ln(\frac{8}{5})}$
• Jan 14th 2009, 05:33 AM
HallsofIvy
Quote:

Originally Posted by mattty
The question is: The bacterial population in a certain culture is given by $\displaystyle P=I(1.6)^t$ where $\displaystyle t$ is in hours and $\displaystyle I$ is the initial population (when $\displaystyle t=0$). Determine how long it takes for this population to double in size.

So, from this, i've worked out that $\displaystyle P=2I$ and then i took a stab at rearranging the equation into the form $\displaystyle x={log_by}$ and got
$\displaystyle t=I\log_{1.6}2I$

Your equation is, then, $\displaystyle 2I= I(1.6)^t$. The very first thing you should see is that the "I"s cancel, giving $$2= (1.6)^t[/quote] so that your answer does not depend on I. Taking the log, base 1.56 would give \displaystyle t= log_{1.6}(2) but, for some reason, they forgot to put a "\displaystyle log_{1.6}" key on my calculator! Fortunately, logarithms are pretty much interchangable. What do you get if you take ln of both sides? That is, \displaystyle ln(2)= ln(1.6^t). Use a "law of logarithms" to reduce the right side. Quote: Now, although I believe this to be the right answer something isn't ringing true since natural logarithms didn't come into it at all. I've always been told that population growth was one of the main things natural logs were for outside of calculus. Any help or suggestions would be greatly appreciated. • Jan 14th 2009, 05:34 AM Mush Quote: Originally Posted by mattty The question is: The bacterial population in a certain culture is given by \displaystyle P=I(1.6)^t where \displaystyle t is in hours and \displaystyle I is the initial population (when \displaystyle t=0). Determine how long it takes for this population to double in size. So, from this, i've worked out that \displaystyle P=2I and then i took a stab at rearranging the equation into the form \displaystyle x={log_by} and got \displaystyle t=I\log_{1.6}2I Now, although I believe this to be the right answer something isn't ringing true since natural logarithms didn't come into it at all. I've always been told that population growth was one of the main things natural logs were for outside of calculus. Any help or suggestions would be greatly appreciated. You can you logarithms to ANY base for this problem. The reason for using logarithms to solve this problem is that the laws of logarithms allow us to linearise the equation so it can be solved using simple algebra. These laws of logarithms hold for logarithms of ANY base. So whether you use \displaystyle \ln{} or \displaystyle \log_{2453465345552352345}{} doesn't really matter. Choose to use natural logarithms if you please: \displaystyle P = 2I  \displaystyle \rightarrow I(1.6)^t= 2I  \displaystyle \rightarrow (1.6)^t= 2  \displaystyle \rightarrow \ln{(1.6)^t}= \ln{2}  \displaystyle \rightarrow t \ln{(1.6)}= \ln{2}  \displaystyle \therefore t = \frac{\ln{2}}{\ln{(1.6)}}  • Jan 14th 2009, 06:18 AM mattty Quote: Originally Posted by galactus Don't over complicate it. It's very much my worst trait, i always tend to overcomplicate everything unfortunately. Quote: Originally Posted by HallsofIvy Your equation is, then, \displaystyle 2I= I(1.6)^t. The very first thing you should see is that the "I"s cancel, giving [itex]2= (1.6)^t[/quote ] so that your answer does not depend on I. Taking the log, base 1.56 would give \displaystyle t= log_{1.6}(2) but, for some reason, they forgot to put a "\displaystyle log_{1.6}" key on my calculator! I guess I was just too eager to convert to the [math x={log_by}$$ form and didn't notice, feeling a bit foolish now(Headbang)
on the upside though, at least my calculator does come with a "$\displaystyle log_{1.6}$" key, gotta love the ti nspires :)

many thanks to all 3 of you anyway(Yes)(Yes)
• Jan 14th 2009, 06:29 AM
galactus
Wow, the TI inspire has a built-in log base change?. That's cool.

On all calculators I have seen, even my beloved Voyage 200, one still has to change bases if one wants a base other than 10.

At least, I think so. Maybe I don't know what I'm doing.(Thinking)
• Jan 14th 2009, 06:47 AM
mattty
yep, it doesn't even default to base 10, when you press the log button you automatically need to put in the base and x values yourself. the nspire is great for working with algebra too, if only i could figure out how to set domains and ranges for the graphs i would be estatic -_-
• Jan 14th 2009, 07:30 AM
Mush
Quote:

Originally Posted by mattty
yep, it doesn't even default to base 10, when you press the log button you automatically need to put in the base and x values yourself. the nspire is great for working with algebra too, if only i could figure out how to set domains and ranges for the graphs i would be estatic -_-

My Casio fx-570ES does that. It's not big deal!
• Jan 14th 2009, 12:28 PM
mr fantastic
Quote:

Originally Posted by galactus
Wow, the TI inspire has a built-in log base change?. That's cool.

On all calculators I have seen, even my beloved Voyage 200, one still has to change bases if one wants a base other than 10.

At least, I think so. Maybe I don't know what I'm doing.(Thinking)

Depending on what OS version you're running, the TI-89 does it too.