The subtraction theorem states that for all real numbers,

and

,

.
(To subtract, we can add the inverse.)
Thus, we can have the these two equivalent expressions.
To calculate the remaining caffeine, we use the radioactive decay formula which is expressed as An = Aoe^-kt where An is the amount left after t time, Ao is the initial amount and k is a constant we can calculate from the half-life information. We do as follows:
at half-life,
ln 1/2 = -k(6)
k = 0.12/hr
An = 80e^-0.12(14)
An = 15.87 mg
X=3,-1 your answer is right here
Answer: $48.93
Step-by-step explanation: 789.18 x 0.062