Logarithmic Functions. By: Melissa Meireles & Noelle Alegado. Section 5.1 The Natural Logarithmic Function: Differentiation. The natural logarthmic function is defined by: ln x = ∫ 1/t dt x > 0 The natural logarithmic function has the following properties:
By: Melissa Meireles & Noelle Alegado
f(g(x)) = x for each x in the domain of g
g(f(x)) = x for each x in the domain of f.
The function g is denoted by fˉ¹ (read “f inverse”).
» Reflective Property: The graph of f contains the point (a,b) if and only if the graph of fˉ¹ contains the point (b,a).
» The horizontal line test: states that a function that a function f has an inverse function if and only if every horizontal line intersects the graph of f at most once.
» Guidelines for Finding an Inverse Function
Let f be a function whose domain is an interval I. If f has an inverse function, then the following statements are true:
Let f be a function that is differentiable on an interval I. If f has an inverse function g, then g is differentiable at any x for which f ‘ (g(x)) ≠ 0. Moreover,
g’(x) = 1/(f ’(g(x))). f ’(g(x)) ≠ 0.
x -> -∞x -> ∞
If a is a positive real number (a≠1) and x is any real number, then the exponential function to the base a is denoted by a^x and is defined by
a^x = e^(ln a)x.
If a = 1, then y = 1^x is a constant function.
If a is a positive real number and x is any real number, then the exponential function to the base a is denoted as
logaX = (1/lna)(lnx).
»The common logarithmic function is the log function to the base 10.
Let a be a positive real number (a ≠ 1) and let u be a differentiable function of x.
Let n be any real number and let u be a differentiable function of x.