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Logarithmic Functions

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Logarithmic Functions

By: Melissa Meireles & Noelle Alegado

- 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.

- 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

- 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)

- 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

- 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.

» 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.

- 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).

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.

- 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

- 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 -> ∞

- 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)

- Let u be a differentiable function of x.
- 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

- Let u be a differentiable function of x.

- 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.

- 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

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