A population of bacteria is growing according to the function below, where t is in hours. How many hours will it take for the population to grow to 12,000 bacteria? Round your answer to the nearest integer. Do not include units in your answer.

Question

campbell_st · Answer

do you have the function...?

whpalmer4 · Answer

@campbell_st where would be the sport in that?! :-)

campbell_st · Answer

I know... I could come up with a nice exponential model... but the value of the constant is needed...

anonymous · Answer

Yeah I have it.. sorry
B(t) = 50  e2t

terenzreignz · Answer

$$\Large B(t) = 50e^{2t}$$
...models how many bacteria after t hours, right?
So, when the number of bacteria is 12000, then...
$$\Large 12000 = 50e^{2t}$$
Can you solve for t?

anonymous · Answer

240/e^2? I don't know how to reduce it much more than that..

terenzreignz · Answer

Oh... no... I see you divided both sides by 50... everything's fine to that point...
$$\Large 240 = e^{2t}$$
However, at this point, one does not simply separate the base from exponents... you need the help of the 'natural logarithm' function :)

whpalmer4 · Answer

No, you have to take the natural log of both sides, giving you $$\ln 240=2t$$$$\frac{\ln 240}{2} = t$$and then break out the calculator or your memory of various values of $\ln$

terenzreignz · Answer

Make use of the fact that
$$\Large \ln e^{x}=x$$

anonymous · Answer

I'm sorry, I really have no idea what it means :/ I take an online course that gives you an answer, but not a way to actually figure out the question..

terenzreignz · Answer

Right. Using this fact...
$$\Large \color{blue}\ln \color{red}e^{x} = x$$(ln and the exponential function cancel each other out...)

And $$\Large 240 = e^{2t}$$

You can take the ln of both sides and they'd still be equal...
$$\Large \ln 240 = \ln e^{2t}$$

Using the property above, what becomes of the right-side?

anonymous · Answer

I have no idea.. I'm supposed to leave tomorrow and i'm on the final question of my final exam, I could really just use the help :/

terenzreignz · Answer

See the analogue...
$$\Large \ln e^{\color{red}x}=\color{red}x$$$$\Large \ln e^{\color{red}{2t}}=\color{green}?$$

anonymous · Answer

so.. In e^2t = In240. 
In e^x = x = In e^2t
Which makes the whole equation come out to 240?

anonymous · Answer

or would 2t = 240?

terenzreignz · Answer

Slowly... I just pointed out that the ln of e raised to something... is that something...
$$\Large \ln e^{x}=x$$
Look~ the ln and the e just disappear, leaving whatever it was that e was raised to.

terenzreignz · Answer

On the other hand, the ln does not 'disappear' in ln 240 because it's not $$\Large \ln e^{240}$$

anonymous · Answer

Okay, In e^(240) = 240?

terenzreignz · Answer

Yes, that is true, but that is irrelevant in this question.
However, using that idea, what is 
$$\Large \ln e^{2t}=\color{red}?$$

anonymous · Answer

Okay, so In e^(2t) = In e^240)?

anonymous · Answer

which would make the question equal to 120?

terenzreignz · Answer

There is no $$\Large \ln e^{240}$$ involved.
I just put it up for the sake of illustrating that...
$$\Large \ln 240 \ne 240$$$$\Large \ln e^{240}= 240$$

anonymous · Answer

Oh okay, one moment then..

anonymous · Answer

2.7403?

anonymous · Answer

I did 240 = e^(2t) on a calculator, and that's what it gave me :3

terenzreignz · Answer

But that's cheating -.-
Well, anyway, given that this can't be solved without a calculator, it's correct.

whpalmer4 · Answer

Well, I wouldn't go so far as to say it can't be solved without a calculator.  If you've memorized the values of $\ln 4, \ln 6, \ln 10$, add them up and divide by 2.  I find it rather useful to know the first dozen logs and square roots...