Question 2. Logs and Growth Rates. Recall the definition of the natural logarithm: We say that log(x) =y if x =ey , where e =2.718 ... is a known constant. For this course, log and In are the same things. Taking the log of a variable can provide a useful data transformation. To treat data, it might help just to think of the log function as a particular function with some appealing properties, which we will explore in the questions below. Recall the following property of exponents and logarithms: axa = a abtc log(x x y) = log(x) + log(y), log(x") = a x log(a), log(er) = x. a) Go to https://www.desmos.com/calculator. Draw the graph of In(1 + x) and x . When do you think In(1 + x) is a good approximation to x ? Nevertheless, in this class, we always assume that it is a good approximation. b) Show that gowth rate can be approximated by log at - 1c) Let Z, 2X, XY, for t: 1,2,3,--- . Let g,- be the growth rate of variable 2' , where i can be Z ,Y,X .. I frequently use this shortcut, instead of dening the growth rate for each variable, I say that this, which means the growth rate of Z is gz , of Y is gy and X is 9;; . Show that the growth rate of Z can be approximated by 9X + gy . d) Go to https://fred.stlouisfed.org/ and search for \"Real GDP\". From \"edit graph\