Both antibiotics were prescribed in high dosage slow release capsules. The function
C(x) = 5log(x +1) +10 models the concentration of levofloxacin in mol/L over a time x, in hours. The
function D(x) =10log(x+1)+5 models the concentration of metronidazole in mol/L over a time x, in hours. Which of the two drugs has a higher initial concentration in the blood stream? Justify your answer
with an explanation.
To determine which drug has a higher initial concentration in the bloodstream, we need to compare the values of C(0) and D(0).
For C(x), the concentration of levofloxacin, we have:
C(0) = 5log(0 + 1) + 10 = 5log(1) + 10 = 5(0) + 10 = 10 mol/L
For D(x), the concentration of metronidazole, we have:
D(0) = 10log(0 + 1) + 5 = 10log(1) + 5 = 10(0) + 5 = 5 mol/L
Comparing these values, we can see that C(0) = 10 mol/L is higher than D(0) = 5 mol/L. Therefore, the drug levofloxacin has a higher initial concentration in the bloodstream compared to metronidazole.
To determine which drug has a higher initial concentration, we need to compare the values of C(0) and D(0) since initial concentration is when x = 0.
For the function C(x) = 5log(x + 1) + 10, we substitute x = 0:
C(0) = 5log(0 + 1) + 10
= 5log(1) + 10
= 5(0) + 10
= 0 + 10
= 10
For the function D(x) = 10log(x + 1) + 5, we substitute x = 0:
D(0) = 10log(0 + 1) + 5
= 10log(1) + 5
= 10(0) + 5
= 0 + 5
= 5
Comparing the values, we see that C(0) = 10 and D(0) = 5. This means that the initial concentration of levofloxacin (C(x)) is higher than the initial concentration of metronidazole (D(x)) in the blood stream.