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.
Determine when C(x) = D(x) algebraically and state what this represents in this situation.
To determine which drug has a higher initial concentration in the bloodstream, we need to compare the initial values of the functions C(x) and D(x).
For C(x) = 5log(x + 1) + 10:
The initial value can be found by plugging in x = 0:
C(0) = 5log(0 + 1) + 10 = 5log(1) + 10 = 5(0) + 10 = 10 mol/L
For D(x) = 10log(x + 1) + 5:
The initial value can also be found by plugging in x = 0:
D(0) = 10log(0 + 1) + 5 = 10log(1) + 5 = 10(0) + 5 = 5 mol/L
Comparing the initial concentrations, we can see that the initial concentration of levofloxacin (C(x)) is 10 mol/L, while the initial concentration of metronidazole (D(x)) is 5 mol/L.
Therefore, the initial concentration of levofloxacin is higher than the initial concentration of metronidazole in the bloodstream.
To find when C(x) = D(x) algebraically, we need to solve the equation:
5log(x + 1) + 10 = 10log(x + 1) + 5
Subtracting 5log(x + 1) from both sides, we get:
10 = 5log(x + 1)
Dividing both sides by 5, we have:
2 = log(x + 1)
To convert to exponential form, we rewrite the equation as:
10^2 = x + 1
Simplifying, we find:
100 = x + 1
Subtracting 1 from both sides, we get:
x = 99
Therefore, when x = 99 hours, the concentrations of levofloxacin and metronidazole are equal.
To determine which drug has a higher initial concentration in the bloodstream, we need to compare the values of C(x) and D(x) at x = 0.
For C(x):
C(x) = 5log(x + 1) + 10
Substitute x = 0:
C(0) = 5log(0 + 1) + 10
C(0) = 5log(1) + 10
C(0) = 5(0) + 10
C(0) = 10
For D(x):
D(x) = 10log(x + 1) + 5
Substitute x = 0:
D(0) = 10log(0 + 1) + 5
D(0) = 10log(1) + 5
D(0) = 10(0) + 5
D(0) = 5
Comparing the values, we see that C(0) = 10 and D(0) = 5. Therefore, the initial concentration of levofloxacin (C(x)) is higher in the bloodstream than the initial concentration of metronidazole (D(x)).
To determine when C(x) = D(x) algebraically, we set the two functions equal to each other:
5log(x + 1) + 10 = 10log(x + 1) + 5
Subtracting 5log(x + 1) from both sides:
10 = 5log(x + 1)
Dividing both sides by 5:
2 = log(x + 1)
Taking the antilogarithm of both sides (base 10):
x + 1 = 10^2
x + 1 = 100
Subtracting 1 from both sides:
x = 99
So, when x = 99 hours, the concentrations of levofloxacin and metronidazole are equal. In this situation, it represents the time at which the concentrations of both drugs are the same in the bloodstream.