Logarithmic Functions

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# Logarithmic Functions - PowerPoint PPT Presentation

Logarithmic Functions. By: Melissa Meireles &amp; Noelle Alegado. Section 5.1 The Natural Logarithmic Function: Differentiation. The natural logarthmic function is defined by: ln x = ∫ 1/t dt x &gt; 0 The natural logarithmic function has the following properties:

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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:
• 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
• 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 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
• 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
• 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…

» 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
• 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
• 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 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
• 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
• 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: