Logarithmic Functions
By: Melissa Meireles & Noelle
Alegado
Section 5.1 The Natural Logarithmic
Function: Differentiation
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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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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
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Logarithmic Properties: If a and b are
positive numbers and n is rational, then
the following properties are true.
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ln
ln
ln
ln
(1) = 0
(ab) = ln a + ln b
(a^n) = n ln a
(a/b) = ln a – ln b
5.1 Continued
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Definition of e: The letter e denotes the
positive real number such that
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Derivative of the Natural Logarithmic
Function: Let u be a differentiable function
of x
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ln e = ∫ (1/t) dt = 1
(d/dx) [ln x] = (1/x), x > 0
(d/dx) [ln u] = (1/u) (du/dx) = (u`/u), u > 0
Derivative Involving Absolute Value
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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:
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1.
2.
∫1/x dx = ln|x|+C
∫1/u du = ln|u|+C
Integrals for Trigonometric Functions
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∫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
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And
f(g(x)) = x for each x in the domain of g
g(f(x)) = x for each x in the domain of f.
The function g is denoted by fˉ¹ (read “f inverse”).
Important Observations:

1.
2.
3.
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).
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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.
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Theorem 5.7:
1.
A function has an inverse function if and only if it is one-to-one.
2.
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
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Derivative of an Inverse Function
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1.
2.
3.
4.
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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
If
If
If
If
Let f be a function whose domain is an interval I. If f has an inverse function, then the following
statements are true:
f is continuous on its domain, the fˉ¹ is continuous on its domain.
f is increasing on its domain, then fˉ¹ is increasing on its domain.
f is decreasing on its domain, then fˉ¹ is decreasing on its domain.
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
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Definition of the Natural Exponential Function
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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
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Let a and b be any real numbers.
e^ae^b = e^a+b
e^a/e^b = e^a-b
5.4 Continued
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Properties of the Natural Exponential
Function
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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
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The Derivative of the Natural Exponential
Function
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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
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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.
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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).
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Properties of Inverse Functions
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1.
2.
3.
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.
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1.
2.
3.
4.
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.
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1.
2.
d/dx [x^n] = nx^n-1
d/dx [u^n] = nu^n-1 du/dx
For further explanations:
1.
2.
3.
http://youtube.com/watch?v=W6kLsUU53w
http://youtube.com/watch?v=YO9MymFe
u40&feature=related
http://youtube.com/watch?v=nxsPlxYbLJ
4&feature=related