Logarithmic functions
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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

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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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