Question

I am stuck on an exponential decay/log question. Question and work attached. I have no idea how to go further in my work.

Answer 1

no wonder you are stuck
they are using an \(a\) instead of \(e\) which is what you were trying to solve
\[B=B_0e^{kt}\] but they only asked for
\[B=B_0a^{kt}\] which actually makes it much easier

Answer 2

if you want to use any base, rather than \(e\) you can use \(\frac{200}{500}=\frac{2}{5}\) and write your formula as
\[B=500\left(\frac{2}{5}\right)^{\frac{t}{2}}\] as your population decreases to \(\frac{2}{5}\) of its original amount every two hours

Answer 3

you only need to find the \(k\) using the log as you did, if you are asked to find the equation in the form
\[B=B_0e^{kt}\] in which case \(B_0=500\) and you solve for \(k\) via
\[\ln(\frac{2}{5})=2k\] making
\[k=\frac{\ln(\frac{2}{5})}{2}\]

Answer 4

Oh ok. that makes it easier. I was having a hard time solving it since there was two unknowns and I got confused. Thanks for the explanation.