Question

A pharmaceutical scientist studying two medications wonders how long different amounts of each medicine stay in someone's bloodstream. The amount of time (in hours) one medication stays in the bloodstream can be modeled by f(x)=-1.25\cdot \ln{\left(\dfrac{1}{x}\right)}f(x)=−1.25⋅ln( x 1 ​ )f, left parenthesis, x, right parenthesis, equals, minus, 1, point, 25, dot, natural log, left parenthesis, start fraction, 1, divided by, x, end fraction, right parenthesis, where xxx is the initial amount of the medicine (in milligrams). The corresponding function for the other medication is g(x)=-1.8\cdot\ln\left(\dfrac{2.1}{x}\right)g(x)=−1.8⋅ln( x 2.1 ​ )g, left parenthesis, x, right parenthesis, equals, minus, 1, point, 8, dot, natural log, left parenthesis, start fraction, 2, point, 1, divided by, x, end fraction, right parenthesis. Here are the graphs of

Answer 1

The question pertains to the use of logarithmic functions to model drug concentration over time in pharmacokinetics. The functions provided describe how long two different medications stay in the bloodstream based on their initial amounts. This relates to understanding drug half-life and dosing intervals.

Step-by-step explanation:

The student is asking about the mathematical modeling of medication concentration in the bloodstream over time using logarithmic functions. Specifically, the functions given are f(x) = -1.25 · ln(1/x) and g(x) = -1.8 · ln(2.1/x), where x is the initial amount of medicine in milligrams. These functions are likely derived from the pharmacokinetics equation that describes the decay of drug concentration in the body which can be expressed as amount remaining = (amount initial) × e⁻, where e is the base of natural logarithms, t is the elapsed time, and t1/2 is the half-life of the drug. The half-life is significant because it indicates how often the drug must be administered to maintain its therapeutic effect. It is also vital to understand drug metabolism and multiple dosing for drugs with a short half-life versus a single dosing for drugs with a long half-life.

Answer 2

It gives the initial amount + It gives the solution

Step-by-step explanation:

It gives the initial amount + It gives the solution

We know from the graph given and the data provided that the function can automatically be proven true.

It cannot describe the amount of time as the data says X is how much of the medicine is given, not the time.

It can't be the equal ratio because, for example, 1 milligram of medicine doesn't equal 1h of time in the body.

All you need to do for this is educational guessing, reasoning, and the process of elimination. Hope this helped! have a great day :7)