$($c$)$ Derive $ln(\frac{L(t)}{L_0})=-kt$ for a BOD test. Start your answer by defining $L(t)=BO{D}_L-$BOD$(t).$ Assume first$-$order decay of $L$ during the BOD test $($i$.$e$.,d\frac{L}{d}t=-kL).$

$($d$)$ BOD test results are given below, plot $-ln(\frac{L(t)}{L_0})$ vs$.$ time. You will have to guess a value for $BO{D}_L.$ Assume $BO{D}_L$ is $334mg-\frac{O_2}{L}.$ Find the first order kinetic constant $k$ from the plot. $($If you use MS Excel, right click on plotted data points, choose "Add Trendline," check for "Linear," set $"0"$ for "Set Interception," and check for "Display equation" and "Display R$-$square value."$)$ Show the resulting plot in your answer. Once $k$ is found, calculate $BO{D}_L$ and compare it with the initial guess.

$\backslash$table$[[$Incubation time $($days$),0,1,2,3,4,5],[$Measured BOD $(mg\frac{O_2}{L}),0,66,109,147,180,208]]$

$($e$)$ If the initial guess of $BO{D}_L(334m\frac{g}{L})$ were not given, write a short discussion on how you would find $BOD{L}_L.$