3. (a) Without the use of a calculator find either an exact value or a range of values of the form (a, a + 1) for the following quantities. (i) log:43 (ii) log,156 (ii) log 2,100,000 (iv) log3 (-log2) [b) Consider the function f (x) in Problem 1b of this assignment. Find its inverse f-1(x) and graph both the function and its inverse on the same coordinate system. Also, compare the domain and range of / (x) to that of f=1(x). Finally, comment on the relationship between the graphs of the two functions about the line y = x.4. (a] Solve the following equation. Clearly explain whether all solutions are acceptable. 72x + 6x-5 = 343 [b) Solve the following equation. Clearly explain whether all solutions are acceptable. + 2 (c) Solve the following equation. Clearly explain whether all solutions are acceptable. [Hint: Use properties of logarithms to condense the expression and write it as a single logarithm of a function with coefficient 1. What is the domain of the function?] logz (x* + 2x2 - x - 2) - log, (x + 1) - 2 log=(x3 + x - 2) = -71082 x + 15. (2) A species is in danger of extinction; its population is decreasing exponentially as P = Piper, where P. Po, & are, respectively, the current population, the original population (at timer = 0], and the rate of decay in appropriate units. (1 If 10 years ago we had P = 1500, and today we have P = 1000, find the decay rate. What does its sign signify? (in) if the population of the species drops below 100, the situation wil be incversible. When will this happen? (b] The radioactive element potassium 40 decays exponentially with a half-life of 1.31 billion years. (0 Find its decay rate. (i) The age of a dinosaur discovered in 1964 was estimated using potassium-40 dating of rocks surrounding the bones. The analysis indicated that 945% of the original amount of potassium-40 was still present. Estimate the age of the bones of the dinosaur. (c) Exponential growth is possible only when the available natural resources cannot be exhausted, which is not true in the real world. Charles Darwin recognized this fact; in the "struggle for existence", he stated that individuals will compete [with members of their own or other species] for limited resources. The successful ones will survive to pass on their own characteristics and traits to the next generation at a greater rate, a process known as natural selection. To model the reality of limited resources, population ecologists developed the logistic growth model In the model, while the original growth has an exponential behavior, this growth will level off as resources become scarce. The population sing at this plateau represents the maximum population size that a particular environment can support; it is called the carrying capacity (KJ. In other words, the carrying capacity of an organism in a given environment is defined to be the maximum population of the organism that the environment can sustain indefinitely. 7