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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:

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logarithmic functions

Logarithmic Functions

By: Melissa Meireles & Noelle Alegado

section 5 1 the natural logarithmic function differentiation
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:
    • The domain is (0,∞) and the range is (-∞,∞).
    • The function is continuous, increasing, and one-to-one.
    • The graph is concave downward.
5 1 continued
5.1 Continued
  • Logarithmic Properties: If a and b are positive numbers and n is rational, then the following properties are true.
    • ln (1) = 0
    • ln (ab) = ln a + ln b
    • ln (a^n) = n ln a
    • ln (a/b) = ln a – ln b
5 1 continued1
5.1 Continued
  • Definition of e: The letter e denotes the positive real number such that
    • ln e = ∫ (1/t) dt = 1
  • Derivative of the Natural Logarithmic Function: Let u be a differentiable function of x
    • (d/dx) [ln x] = (1/x), x > 0
    • (d/dx) [ln u] = (1/u) (du/dx) = (u`/u), u > 0
  • Derivative Involving Absolute Value
    • If u is a differentiable function of x such that u ≠ 0, then (d/dx) [ln lul] = (u`/u)
5 2 the natural logarithmic function integration
5.2-The Natural Logarithmic Function: Integration
  • Log Rule for differentiation: d/dx [ ln|x| ] = 1/x AND d/dx [ ln|u| ] = u’/u
  • Log rule for integration:
    • ∫1/x dx = ln|x|+C
    • ∫1/u du = ln|u|+C
  • Integrals for Trigonometric Functions
    • ∫sinu du = -cosu+C
    • ∫cosu du= sinu+C
    • ∫tanu du= -ln|cosu|+C
    • ∫cotu du= ln|sinu|+C
    • ∫secu du= ln|secu+tanu|+C
    • ∫cscu du= -ln|cscu+cotu|+C
5 3 inverse functions
5.3-Inverse Functions
  • A function g is the inverse function of the function f if

f(g(x)) = x for each x in the domain of g

And

g(f(x)) = x for each x in the domain of f.

The function g is denoted by fˉ¹ (read “f inverse”).

  • Important Observations:
    • If g is the inverse function of f, then f is the inverse of g.
    • The domain of fˉ¹ is equal to the range of f, and the range of fˉ¹ is equal to the domain of f.
    • A function need not have an inverse function, but if it does, the inverse function is unique.

» Reflective Property: The graph of f contains the point (a,b) if and only if the graph of fˉ¹ contains the point (b,a).

  • Existence of an Inverse Function:

» 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.

    • Theorem 5.7:
    • A function has an inverse function if and only if it is one-to-one.
    • If f is strictly monotonic on its entire domain, then it is one-to-one and therefore has an inverse function.
section 5 3 con t
Section 5.3 Con’t…

» Guidelines for Finding an Inverse Function

    • Use Theorem 5.7 to determine whether the function given by y= f(x) has an inverse function.
    • Solve for x as a function of y: x = g(y) = fˉ¹(y).
    • Interchange x and y. The resulting equation is y = fˉ¹(x).
    • Define the domain of fˉ¹ to be the range of f.
    • Verify that f (fˉ¹(x))= x and fˉ¹(f(x))=x.
  • Derivative of an Inverse Function
    • Continuity and Differentiability of Inverse Functions

Let f be a function whose domain is an interval I. If f has an inverse function, then the following statements are true:

    • If f is continuous on its domain, the fˉ¹ is continuous on its domain.
    • If f is increasing on its domain, then fˉ¹ is increasing on its domain.
    • If f is decreasing on its domain, then fˉ¹ is decreasing on its domain.
    • If f is differentiable at c and f ‘ (c) ≠ 0, then fˉ¹ is differentiable at f(c).
  • The derivative of an Inverse Function

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.

5 4 exponential functions differentiation and integration
5.4 Exponential Functions: Differentiation and Integration
  • Definition of the Natural Exponential Function
    • The inverse function of the natural logarithmic function f(x) = ln x is called the natural exponential function and is denoted by f^-1(x) = e^x That is, y = e^x if and only if x = ln y
  • Operations with Exponential Functions
    • Let a and b be any real numbers.
    • e^ae^b = e^a+b
    • e^a/e^b = e^a-b
5 4 continued
5.4 Continued
  • Properties of the Natural Exponential Function
    • The domain of f(x) = e^x is (-∞,∞), and the range is (0,∞).
    • The function f(x) = e^x is continuous, increasing, and one-to-one on its entire domain.
    • The graph of f(x) = e^x is concave upward on its entire domain.
    • Lim e^x = 0 and lim e^x = ∞

x -> -∞x -> ∞

5 4 continued1
5.4 Continued
  • The Derivative of the Natural Exponential Function
    • Let u be a differentiable function of x.
      • (d/dx) [e^x] = e^x
      • (d/dx) [e^u] = e^u (du/dx)
  • Integration Rules for Exponential Functions
    • Let u be a differentiable function of x.
      • ∫e^x dx = e^x + c
      • ∫e^u dx = e^u + c
5 5 bases other than e and applications
5.5-Bases Other than e and Applications
  • Definition of Exponential Function to Base a

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.

  • Definition of Logarithmic Function to Base a

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).

  • Properties of Inverse Functions
    • Y=a^x if and only if x=logaY
    • a^logaX=X, for x>0.
    • Loga a^x = x. for all x.

»The common logarithmic function is the log function to the base 10.

5 5 con t differentiation and integration
5.5 Con’t…-Differentiation and Integration
  • Derivatives for Bases Other than e

Let a be a positive real number (a ≠ 1) and let u be a differentiable function of x.

    • d/dx [a^x] = (lna)a^x
    • d/dx [a^u] = (lnu)a^u du/dx
    • d/dx [logaX] = 1/(lna)x
    • d/dx [logau] = 1/(lna)u du/dx
  • The Power Rule for Real Exponents

Let n be any real number and let u be a differentiable function of x.

    • d/dx [x^n] = nx^n-1
    • d/dx [u^n] = nu^n-1 du/dx
for further explanations
For further explanations:
  • http://youtube.com/watch?v=W6-kLsUU53w
  • http://youtube.com/watch?v=YO9MymFeu40&feature=related
  • http://youtube.com/watch?v=nxsPlxYbLJ4&feature=related
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