Thread: Word Problem w/exponents

The antibiotic clarithromycin is eliminated from the body according to the formula:
A(t)=500e^-0.1386t
A is amount remaining of mg remaining in body after (t) hours the drug reaches peak concentration. How much time will pass before the amoun of drug in the body is reduced to 100mg?

This is as far as I've gotten....I don't think I'm even close.
100(t)=500e^(-0.1386(t))
-100 -100
__________________
t=400e^(0.1386(t))
Thanks for any assistance

Hello, crazydaizy78!

The antibiotic clarithromycin is eliminated from the body according to the formula:
. . $\displaystyle A(t)\:=\:500e^{-0.1386t}$

$\displaystyle A$ is amount remaining in mg remaining in body after (t) hours.
How much time will pass before the amount of drug in the body is reduced to 100 mg?

You started correctly . . .

We have: .$\displaystyle 100 \:=\:500e^{-0.1386t} $

Divide by 100: .$\displaystyle 1 \;=\;5e^{0.1386t}$

Multiply by $\displaystyle e^{0.1386t}\!:\;\;e^{0.1386t} \;=\;5$

Take logs: .$\displaystyle \ln\left(e^{0.1386t}\right) \;=\;\ln(5)$

And we have: .$\displaystyle 0.1386t\ln(e) \;=\;\ln(5)$

Since $\displaystyle \ln(e) = 1$, we have: .$\displaystyle 0.1386t \;=\;\ln(5)\quad\Rightarrow\quad t \;=\;\frac{\ln(5)}{0.1386} $


. . Therefore: .$\displaystyle t \:\approx\:11.6$ hours.