1 / 13

Logarithmic Functions

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:

newton
Download Presentation

Logarithmic Functions

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Logarithmic Functions By: Melissa Meireles & Noelle Alegado

  2. 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.

  3. 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

  4. 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. 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

  6. 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.

  7. 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.

  8. 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

  9. 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 -> ∞

  10. 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

  11. 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.

  12. 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

  13. 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

More Related