Download Lecture 4. Exponential and Logarithmic Functions.

Lecture 4.
Exponential and Logarithmic Functions.
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Exponential Functions
Denition (Exponential Functions)
If b > 0 and b 6= 1, then there is a unique function called exponential function with
base b that is dened by
f (x ) = b x
for every real number x .
Theorem (Properties of an Exponential Function)
The exponential function f (x ) = bx for b > 0 and b 6= 1 has these properties:
1 It is dened, continuous, and positive f (x ) > 0 for all x ∈ R.
2 The x axis is a horizontal asymptote of the graph of f .
3 The y intercept of the graph is (0, 1); there is no x intercept.
4 If b > 1,
lim bx = 0 and
lim bx = +∞,
x →−∞
If 0 < b < 1,
lim bx = +∞ and
x →−∞
5
x →+∞
lim bx = 0.
x →+∞
For all x, the function is increasing (graph rasing) if b > 1 and decreasing
(graph falling) if 0 < b < 1.
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Exponential Functions
Denition (Euler's Number)
e := 1 +
1
1
1
+
+
+ ...
1 1·2 1·2·3
≈ 2.71828182845904523536028747135266249775724709369995
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Exponential Functions
y
1 x
3
1 x
e
1 x
2
−3
−2
−1
9
8
7
6
5
4
3
2
1
0
3x
ex
2x
1
2
3
x
Graphs of exponential functions
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Logarithmic Functions
Denition (Exponential Functions)
If b > 0, then the logarithm of x to the base b (b > 0, b 6= 1), denoted logb x , is
the number y such that by = x ; that is,
y = logb x if and only if by = x
for x > 0.
Example
log10 1, 000 = 3 since 103 = 1, 000.
log2 32 = 5 since 25 = 32.
log10
1
125
= −3 since 5−3 =
1
125 .
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Logarithmic Functions
Theorem (Logarithmic Rules)
Let b be any logarithmic base (b > 0, b 6= 1). Then
logb 1 = 0 and logb b = 1
and if u and v are any positive numbers, we have
The equality rule
logb u = logb v if and only if u = v
The product rule
logb (uv ) = logb u + logb v
The power rule
logb u r = r · logb u for any real number r
The quotient rule
logb
The inversion rule
logb b = u
u
v
u
= logb u − logb v
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Logarithmic Functions
Theorem (Properties of an Logarithmic Function)
The logarithmic function f (x ) = logb x for b > 0 and b 6= 1 has these properties:
1 It is dened, continuous for all x ∈ R.
2 The y axis is a vertical asymptote of the graph of f .
3 The x intercept of the graph is (1, 0); there is no y intercept.
4 If b > 1,
lim+ logb x = +∞ and
lim logb x = +∞,
x →0
x →+∞
If 0 < b < 1,
lim bx = +∞ and
x →0+
5
lim logb x = +∞.
x →+∞
For all x > 0, the function is increasing (graph rasing) if b > 1 and decreasing
(graph falling) if 0 < b < 1.
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Exponential Functions
Denition (Natural Logarithm)
The logarithm loge x is called the natural logarithm of x and is denoted by ln x that
is, for x > 0
y = ln x if and only if e y = x
Theorem (Conversion Formula for Logarithms)
If a and b are positive numbers with b 6= 1, then
logb a =
ln a
.
ln b
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Logarithmic Functions
y
log2 x
3
lnx
log3 x
2
1
0 1 2 3 4 5 6 7 8 9
x
−1
−2
−3
log 13 x
log e1 x
log 12 x
Graphs of logarithmic functions
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Dierentiation of Exponential and Logarithmic Functions
Theorem (The Derivative of e x )
For every real number x
(e x )0 = e x
Theorem (The Chain Rule for e u )
If u (x ) is a dierentiable function of x, then
e u (x )
0
= e u(x ) · u 0 (x ).
Theorem (The Derivative of ln x )
For all x > 0
(ln x )0 =
1
x
Theorem (The Chain Rule for ln u )
If u (x ) is a dierentiable function of x, then
(ln u (x ))0 =
u 0 (x )
for u (x ) > 0.
u (x )
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Thank you for your attention!
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