Logarithmic and Exponential Functions-Questions and Answers Graded A+

Logarithmic and Exponential Functions-Questions and Answers Graded A+ Definition one to one function - ANSWER-A function f with domain D and range R is one to one if whenever a≠b in D, then f(a)≠f(b) in R. Definition Inverse Function - ANSWER-Let f be a one to one function, a function g with domain R and range D is the inverse of f if y = f(x) and x = g(y) Guidelines for Finding f⁻¹ - ANSWER-Verify that f is one to one, solve y=f(x) for x in terms of y, obtaining x = f⁻¹(y). Theorem Differentiable Inverse Function - ANSWER-g'(c) = 1/f'(g(c)) Corollary Differentiable Inverse Function - ANSWER-g'(x) = 1/f'(g(x)) Definition of Natural Logarithmic Function - ANSWER-ln x = ∫t→x (1/t) dt Theorem Dx ln(x) - ANSWER-1/x Theorem Dx ln(u) for u =g(x) - ANSWER-(1/u)Dx u Law of Natural Log ln(pq) - ANSWER-ln(p)+ln(q) Law of Natural Log ln(p/q) - ANSWER-ln(p)-ln(q) Law of Natural Log ln(p)ⁿ - ANSWER-n ln(p) Guidelines for Logarithmic Differentiation - ANSWER-start with given function, take natural log and simplify, differentiate implicitly, multiply by y=f(x) Definition natural exponential function - ANSWER-the inverse of the natural logarithmic function Definition of number e - ANSWER-the letter e denotes the positive real number such that ln(e) = 1 Approximation of e - ANSWER-≈ 2.71828 Definition of eⁿ - ANSWER-eⁿ = y if ln(y) = n Theorem ln(e)ⁿ - ANSWER-= n for every n Theorem e^lnx - ANSWER-= x for every x 0