Question 900913
Your general model in better notation is {{{highlight_green(C(t)=C[o]e^(rt))}}}.
You will want to know the value for r, and you use the half-life fact to find r.
First, solve symbolically for r , and second, use {{{C(t)=(1/2)C[o]}}} with half-life time.


{{{ln(C(t))=ln(C[o])+ln(e^(rt))}}}
{{{rt*1+ln(C[o])=ln(C(t))}}}
{{{rt=ln(C(t))-ln(C[o])}}}
{{{highlight(r=(1/t)ln(C(t)/C[o]))}}}


Substitute the values for half-life:
{{{highlight(r=(1/12.5)ln(1/2))}}}, you would use relative values for the concentrations; knowing {{{C(t)}}} is HALF of {{{C[o]}}}
when the half-life time has passed.


The value for the decay constant, {{{highlight(r=-0.05545)}}};
Notice that r is NEGATIVE.


The decay model for Ambien:  {{{highlight_green(highlight(C(t)=C[o]e^(-0.05545*t)))}}}.

If your standard dose is 10 mg/L, then the model for decay can be specifically written
{{{highlight(highlight(C(t)=10*e^(-.05545t)))}}}.