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) \\ (50)/(100)=e^(-0.17t) \\ 0.5=e^(-0.17t) \\ \ln (0.5)=\ln (e^(-0.17t)) \\ \ln (0.5)=-0.17t \\ (\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.