Understanding the Natural Logarithm Function: Definition, Properties, and Applications - P, Study notes of Calculus

Download Understanding the Natural Logarithm Function: Definition, Properties, and Applications - P and more Study notes Calculus in PDF only on Docsity! •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit 7.1 The Logarithm Defined as an Integral What is ex, ln x? We have seen and used the logarithmic function ln x and exponential function ex many times. But, what are they? How did we describe ex? Intuitive and informal • e is some constant: e = 2.7182821828.... • en when n is an integer; e.g. e2 = e · e etc. • er with r being a rational number, e.g. e 23 = (e2) 13 . • When x is irrational, the precise meaning of ex is not so clear. Then, ln x is defined as the inverse of ex. Also, we claimed without proof that (ex)′ = ex, and (ln x)′ = 1 x . In this chapter, we will give a rigorous approach to the definitions and properties of these functions, and we study a wide range of applied problems in which they play a role. Definition of the Natural Logarithm Function To understand this function, we want to study its • domain •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit • range • graph (monotonicity, concavity etc.) • derivative • inverse • other properties Domain of ln x: x > 0 The function is not defined for x ≤ 0. •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit Example: Find ∫ sec x√ ln (sec x + tan x) dx Familiar algebraic properties of ln x Sketch of proof: Let u (x) = bxr, then d dx ln u (x) = u′ (x) u (x) = brxr−1 bxr = r x .∫ y 1 d dx ln u (x) dy = ∫ y 1 r x dy = r ln y ln u (y)− ln u (1) = r ln y ln (bxr)− ln b = r ln x ln (bxr) = ln b + r ln x The graph and range of ln x • ln x is an increasing function because • the graph of ln x is concave down because •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit • limx→∞ ln x = ∞ because • limx→0+ ln x = −∞ because Thus, the range of ln x is (−∞,∞). The graph of y = ln x looks like The inverse function of ln x and the Number e The function ln x is an increasing function of x with • domain (0,∞); • range (−∞,∞); So, it has an inverse ln−1 x with • domain (−∞,∞); •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit • range (0,∞); The graph of y = ln−1 x = ex is the graph of ln x reflected across the line y = x.