Question

Use logarithms to find the how long it would take for your body to only have 50mg of caffeine remaining in your system. Show all your work.

Answer 1

The function that describes the situation is:

$C(t)=100e^{-0.17t}$

Where C represents the caffeine remaining, and t represents the time.

We want to find t such that C(t)=50. So,

$50=100e^{-0.17t}$

To solve this equation for t, we could use logarithms as follows:

$\begin{gathered} 50=100e^{-0.17t} \\ \frac{50}{100}=e^{-0.17t} \\ 0.5=e^{-0.17t} \\ \ln (0.5)=\ln (e^{-0.17t}) \\ \ln (0.5)=-0.17t \\ \frac{\ln (0.5)}{-0.17}=t \\ 4.08=t \end{gathered}$

Therefore, it would take 4.08 hours for your body to have only 50mg of caffeine remaining.