The original problem What I have tried From the information given in the problem we have the points $(5,360)$ and $(20,1000)$
I found a solution online that used the solution to find k shown on the right of the whiteboard. I know that this is the correct answer because that is the answer in the book and I worked it out another way and got the same answer.
$\frac{360}{1000}=\frac{A_0e^{k5}}{A_0e^{k20}}$
$\frac{9}{25}=e^{-15k}$
$\ln\left(\frac{9}{25}\right)=-15k$
$k=\frac{\ln\left(\frac{9}{25}\right)}{-15}$
I would just like to understand why we are allowed to do that.
While trying to understand I tried what is written on the right in red, but that is not the solution that I got on the right. Can someone please help me understand why that isn't correct
$360=A_0e^{5k}$
$0=A_0e^{5k}-360$
$1000=A_0e^{20k}$
$0=A_0e^{20k}-1000$
$A_0e^{5k}-360=A_0e^{20k}-1000$
$e^{15k}=640$
$\ln(640)=15k$
$k=\frac{\ln(640)}{15}$
You are making fundamental mistakes.
Firstly, you have treated $A_0$ as if it is $1$
If you proceeded to first find $k$, you would get $k= 0.068110$,
and to find $A_0$, you'd solve$A_0*e^{(5*0.068110)}=360 \Longrightarrow A_0 =256$
and you'd find that $A_0e^{5k} = 360, A_0e^{20k} = 1000$
secondly, you can't subtract as $e^{15k} = 1000-360$,
you should revise the laws of exponents