I made a lot of typos the first time i wrote it so i rewrote and edited the question
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)=5log(x+1)+10 models the concentration of levofloxcin and metrinidazole in mol/L over time x, in hours.
a) Which of the two drugs has a higher initial concentration in the blood stream? Justify your answer with an explanation.
b) Determine when C(x) = D(x) algebraically and state what this represents in this situation.
c) If Mathews is instructed to take both antibiotics at once the concentration levels could be modeled by the function (C + D)(x). How would the graph of (C + D)(x) differ from the individual graphs of C(x) and D(x)? Explain.
C(x) = 5log(x+1)+10
D(x)=5log(x+1)+10 These are the same function!
a) To determine which drug has a higher initial concentration in the bloodstream, we need to compare the values of C(x) and D(x) when x = 0.
For C(x), we substitute x = 0 into the equation:
C(0) = 5log(0+1) + 10 = 5log(1) + 10 = 5(0) + 10 = 10
For D(x), we substitute x = 0 into the equation:
D(0) = 5log(0+1) + 10 = 5log(1) + 10 = 5(0) + 10 = 10
Both C(x) and D(x) have the same value of 10 when x = 0. Therefore, both drugs have the same initial concentration in the bloodstream.
b) To determine when C(x) = D(x) algebraically, we need to equate the two equations and solve for x.
5log(x+1) + 10 = 5log(x+1) + 10
We can subtract 5log(x+1) and 10 from both sides of the equation:
0 = 0
This equation is always true, which means C(x) = D(x) for all values of x. In this situation, it represents that the concentrations of both drugs are always equal at any given time.
c) The graph of (C + D)(x) represents the concentration levels of both drugs when taken together.
To understand how the graph of (C + D)(x) differs from the individual graphs of C(x) and D(x), we need to consider the properties of logarithmic functions.
The logarithmic function increases slowly as x increases, meaning the concentration of the drug will also increase at a slower rate. When the drugs are taken together, the concentration levels will be the sum of their individual concentrations at each x value.
Therefore, the graph of (C + D)(x) will have higher concentrations at each x value compared to the individual graphs of C(x) and D(x). It will be shifted upwards and show a faster rate of increase since it represents the combined effect of both drugs.