Please help me answer these question with shown work and explanation so that I can learn from it!!

Graded Problems Your work on the graded problems should demonstrate your best work. Write up thorough solutions that a reader could follow from start to finish. The original problem, your assumptions and reasoning, your mathematical work, and your answer should all be shown and/or explained clearly. See \"Graded Problem Guidelines in the syllabus for more information on expectations and the grading rubric. 1. Solve each equation for the variable, showing all steps a. 200(1.18)' =580 b.iO6[%) =5 log(x+4)+log(x)=2 2. For the function f(x)=2e" -4, find: a. The inverse function. b. All properties of the inverse function (domain, range, asymptotes, intercepts) c. Sketch the inverse function. 3. You go to the doctor and a nurse injects you with 13 milligrams of radioactive dye. After 12 minutes, 4.75 milligrams of dye remain in your system. To leave the doctor's office, you must pass through a radiation detector without sounding the alarm. If the detector will sound the alarm whenever more than 2 milligrams of the dye are in your system, how long will your visit to the doctor take, assuming you were given the dye as soon as you arrived and the amount of dye decays exponentially? 4. A wooden artifact from an archeological dig contains 15 percent of the carbon-14 that is present in living trees. How long ago was the artifact made? (The half-life of carbon-14 is 5730 years)