Recall that f(x) = lnⁿ(x) represents the n-times composition of ln(x), or more explicitly, lnⁿ(…lnⁿ(ln(x))…), composed n times.

For all x > 0, define the function g(x) to be the smallest positive integer such that lnⁿ(x) is less than or equal to 1. In other words, for a given x > 0, g(x) is the minimum number of compositions of ln(x) you need until the value is less than or equal to 1. For example, g(10) = 2 because ln(10) ~= 2.302, and ln²(10) = ln(ln(10)) ~= 0.83 <= 1.

a) What is the range of g(x)?

b) Compute g(1), g(18), g(186). From this result, explain why g(x) does not have an inverse on (0, infinity).

c) Sketch the graph of f(x) on the domain (0, 3814279). Give an explanation for your sketch. Hint: for which value of x is lnⁿ(x) = 1?

IMPORTANT NOTE:

g(x) DOES NOT EQUAL lnⁿ(x)

g(x) = n SO THAT n IS AS SMALL AS POSSIBLE WHEN lnⁿ(x) <= 1