Logarithmic Functions

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