exponential and logarithmic functions.pdf
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Properties of ExponentsExponential FunctionsLogarithmic Functions
Exponential and Logarithmic Functions
University of the Philippines Manila
April 15, 2014
Mathematics 14
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Properties of ExponentsExponential FunctionsLogarithmic Functions
Properties of Exponents
Laws of Exponents
Let a, b , x , y ∈ R1 a
x
ay
= ax +y
2ax
ay = ax −y
3 (ab )x = ax b x
4 ab x
= ax
b x 5 (ax )y = axy
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Properties of ExponentsExponential FunctionsLogarithmic Functions
Properties of Exponents
Theorem
Let a, b , x , y ∈ R and a, b > 0,1 ax is a unique real number
2 a0 = 1
3 if a = 1, then ax = 1
4 a−x = 1
ax
5 if a > 1 with x
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Properties of ExponentsExponential FunctionsLogarithmic Functions
Exponential Functions
DefinitionIf b > 0, b = 1, the exponential function with base b is definedby
f (x ) = b x
for every x ∈ R
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Properties of ExponentsExponential FunctionsLogarithmic Functions
Mathematics 14
P f E
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Properties of ExponentsExponential FunctionsLogarithmic Functions
Properties of the Exponential Function
Let b > 0, b = 1 and f be the exponential function with base b .1 dom f = R
2 ran f = (0, +∞
)
3 x -intercept:none
4 y -intercept: 1
5 If b > 1, then f is increasing.
6 If 0
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Properties of ExponentsExponential FunctionsLogarithmic Functions
The Natural Exponential Function
Euler’s Number
Among all for exponential functions there is one particular basethat plays a special role in Calculus. That base, denoted by the
letter e , is a certain irrational number whose value to six decimalplaces is
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P e ties f E e ts
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Properties of ExponentsExponential FunctionsLogarithmic Functions
The Natural Exponential Function
Euler’s Number
Among all for exponential functions there is one particular basethat plays a special role in Calculus. That base, denoted by the
letter e , is a certain irrational number whose value to six decimalplaces is
e ≈ 2.718282
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Properties of Exponents
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Properties of ExponentsExponential FunctionsLogarithmic Functions
The Natural Exponential Function
Euler’s Number
Among all for exponential functions there is one particular basethat plays a special role in Calculus. That base, denoted by the
letter e , is a certain irrational number whose value to six decimalplaces is
e ≈ 2.718282This base is important in calculus because, b = e is the only base
for which the slope of the tangent line to the curve y = b x at anypoint P on the curve is equal to the y -coordinate at P .
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Properties of Exponents
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Properties of ExponentsExponential FunctionsLogarithmic Functions
Definition
The natural exponential function is the exponential function
with base e :f (x ) = e x .
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Properties of Exponents
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Properties of ExponentsExponential FunctionsLogarithmic Functions
Equations involving Exponential Expressions
Exercises
Solve for the solution set of the following equations:
1 53x = 57x −2
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Properties of Exponents
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Properties of ExponentsExponential FunctionsLogarithmic Functions
Equations involving Exponential Expressions
Exercises
Solve for the solution set of the following equations:
1 53x = 57x −2
2 4t 2
= 23t +2
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Properties of Exponents
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p pExponential FunctionsLogarithmic Functions
Equations involving Exponential Expressions
Exercises
Solve for the solution set of the following equations:
1 53x = 57x −2
2 4t 2
= 23t +2
3 45−9x = 1
8x −2
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Properties of Exponents
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Exponential FunctionsLogarithmic Functions
Equations involving Exponential Expressions
Exercises
Solve for the solution set of the following equations:
1 53x = 57x −2
2 4t 2
= 23t +2
3 45−9x = 1
8x −24
9x
+ 2(3x
) − 3 = 0
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Properties of Exponents
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Exponential FunctionsLogarithmic Functions
Logarithms
DefinitionLet b ∈ R such that b > 0 and b = 1. If b y = x then y is calledthe logarithm of x to the base b , denoted by y = logb x .
Exercises
Let a ∈ R, a > 0 and a = 1.1 log4 16 =
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Properties of ExponentsE i l F i
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Exponential FunctionsLogarithmic Functions
Logarithms
DefinitionLet b ∈ R such that b > 0 and b = 1. If b y = x then y is calledthe logarithm of x to the base b , denoted by y = logb x .
Exercises
Let a ∈ R, a > 0 and a = 1.1 log4 16 =
2 log51
125 =
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Properties of ExponentsE ti l F ti
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Exponential FunctionsLogarithmic Functions
Logarithms
DefinitionLet b ∈ R such that b > 0 and b = 1. If b y = x then y is calledthe logarithm of x to the base b , denoted by y = logb x .
Exercises
Let a ∈ R, a > 0 and a = 1.1 log4 16 =
2 log51
125 =
3 log 1
3
81 =
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Properties of ExponentsExponential Functions
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Exponential FunctionsLogarithmic Functions
Logarithms
DefinitionLet b ∈ R such that b > 0 and b = 1. If b y = x then y is calledthe logarithm of x to the base b , denoted by y = logb x .
Exercises
Let a ∈ R, a > 0 and a = 1.1 log4 16 =
2 log51
125 =
3 log 1
3
81 =
4 loga 1 =
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Properties of ExponentsExponential Functions
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Exponential FunctionsLogarithmic Functions
Logarithms
DefinitionLet b ∈ R such that b > 0 and b = 1. If b y = x then y is calledthe logarithm of x to the base b , denoted by y = logb x .
Exercises
Let a ∈ R, a > 0 and a = 1.1 log4 16 =
2 log51
125 =
3 log 1
3
81 =
4 loga 1 =
5 loga a =
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Properties of ExponentsExponential Functions
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Exponential FunctionsLogarithmic Functions
Relationship of Exponential and Logarithmic Functions
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Properties of ExponentsExponential Functions
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Exponential FunctionsLogarithmic Functions
Relationship of Exponential and Logarithmic Functions
Let b ∈R
such that b > 0 and b = 1.
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Properties of ExponentsExponential Functions
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pLogarithmic Functions
Relationship of Exponential and Logarithmic Functions
Let b ∈R
such that b > 0 and b = 1.Solve for the inverse f −1(x ) of the exponential functionf (x ) = b x .
Interchanging x and y , we have
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Properties of ExponentsExponential Functions
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pLogarithmic Functions
Relationship of Exponential and Logarithmic Functions
Let b ∈R
such that b > 0 and b = 1.Solve for the inverse f −1(x ) of the exponential functionf (x ) = b x .
Interchanging x and y , we have
x = b y
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Properties of ExponentsExponential Functions
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Logarithmic Functions
Relationship of Exponential and Logarithmic Functions
Let b ∈R
such that b >
0 and b = 1.Solve for the inverse f −1(x ) of the exponential functionf (x ) = b x .
Interchanging x and y , we have
x = b y
By definition of a logarithm,
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Properties of ExponentsExponential FunctionsL i h i F i
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Logarithmic Functions
Relationship of Exponential and Logarithmic Functions
Let b ∈R
such that b >
0 and b = 1.Solve for the inverse f −1(x ) of the exponential function
f (x ) = b x .
Interchanging x and y , we have
x = b y
By definition of a logarithm,
logb x = y
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Properties of ExponentsExponential FunctionsL ith i F ti
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Logarithmic Functions
Relationship of Exponential and Logarithmic Functions
Let b ∈R
such that b >
0 and b = 1.Solve for the inverse f −1(x ) of the exponential function
f (x ) = b x .
Interchanging x and y , we have
x = b y
By definition of a logarithm,
logb x = y
Thus,
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Logarithmic Functions
Relationship of Exponential and Logarithmic Functions
Let b ∈R
such that b >
0 and b = 1.Solve for the inverse f −1(x ) of the exponential function
f (x ) = b x .
Interchanging x and y , we have
x = b y
By definition of a logarithm,
logb x = y
Thus,f −1(x ) = logb x
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Logarithmic Functions
Logarithmic Functions
DefinitionLet b ∈ R such that b > 0 and b = 1. The function
f (x ) = logb x
is called the logarithmic function to the base b.
Notes:
1 dom f = (0, +∞)
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Logarithmic Functions
Logarithmic Functions
DefinitionLet b ∈ R such that b > 0 and b = 1. The function
f (x ) = logb x
is called the logarithmic function to the base b.
Notes:
1 dom f = (0, +∞)2 ran f = R
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Properties of ExponentsExponential FunctionsLogarithmic Functions
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Logarithmic Functions
Logarithmic Functions
DefinitionLet b ∈ R such that b > 0 and b = 1. The function
f (x ) = logb x
is called the logarithmic function to the base b.
Notes:
1 dom f = (0, +∞)2 ran f = R
3 logb (b x ) = x for all x ∈ R
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Properties of ExponentsExponential FunctionsLogarithmic Functions
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g u
Logarithmic Functions
DefinitionLet b ∈ R such that b > 0 and b = 1. The function
f (x ) = logb x
is called the logarithmic function to the base b.
Notes:
1 dom f = (0, +∞)2 ran f = R
3 logb (b x ) = x for all x ∈ R
4 b logb x = x for all x ∈ (0, +∞)
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Properties of ExponentsExponential FunctionsLogarithmic Functions
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g
Logarithmic Functions
DefinitionLet b ∈ R such that b > 0 and b = 1. The function
f (x ) = logb x
is called the logarithmic function to the base b.
Notes:
1 dom f = (0, +∞)2 ran f = R
3 logb (b x ) = x for all x ∈ R4 b logb x = x for all x ∈ (0, +∞)5 x -int:1
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Properties of ExponentsExponential FunctionsLogarithmic Functions
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Logarithmic Functions
DefinitionLet b ∈ R such that b > 0 and b = 1. The function
f (x ) = logb x
is called the logarithmic function to the base b.
Notes:
1 dom f = (0, +∞)2 ran f = R
3 logb (b x ) = x for all x ∈ R4 b logb x = x for all x ∈ (0, +∞)5 x -int:1
6 y -int: none
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Logarithmic Functions
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Logarithmic Functions
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Common and Natural Logarithms
Definition
Let x ∈ R such that x > 0.The common logarithm of x, denoted log x , is
log x = log10 x
The natural logarithm of x, denoted ln x , is
ln x = loge x
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Properties of Logarithms
Theorem
If b , x , y > 0, b
= 1, and p
∈R then
1 logb (xy ) = logb x + logb y
2 logb
x
y
= logb x − logb y
3 logb x p = p · logb x
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Exercises
1 Evaluate log2 42014
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Exercises
1 Evaluate log2 42014
2 Evaluate log6 4 + log6 9 − log6 5 − log6 3
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Exercises
1 Evaluate log2 42014
2 Evaluate log6 4 + log6 9 − log6 5 − log6 33 Express log7
7
x 2 + xy
as a sum of constants and
logarithms.
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Exercises
1 Evaluate log2 42014
2 Evaluate log6 4 + log6 9 − log6 5 − log6 33 Express log7
7
x 2 + xy
as a sum of constants and
logarithms.
4 Express 1
2 logb m +
3
2 logb 2n − logb m2n as a single logarithm
with a coefficient 1.
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Exercises
1 Evaluate log2 42014
2 Evaluate log6 4 + log6 9 − log6 5 − log6 33 Express log7
7
x 2 + xy
as a sum of constants and
logarithms.
4 Express 1
2 logb m +
3
2 logb 2n − logb m2n as a single logarithm
with a coefficient 1.
5 Given loga 2 = 0.3 and loga 3 = 0.48. Find loga 72.
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Exercises
1 Evaluate log2 42014
2 Evaluate log6 4 + log6 9 − log6 5 − log6 33 Express log7
7
x 2 + xy
as a sum of constants and
logarithms.
4 Express 1
2 logb m +
3
2 logb 2n − logb m2n as a single logarithm
with a coefficient 1.
5 Given loga 2 = 0.3 and loga 3 = 0.48. Find loga 72.
6 Are f (x ) = log5(x − 2)2 and g (x ) = 2 log5(x − 2) the samefunctions?
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Change of Base Formula
Theorem
If a, b , x ∈ R with a, b , x > 0 and a, b = 1 thenlogb x =
loga x
loga b
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Properties of ExponentsExponential FunctionsLogarithmic Functions
E i i l i L i h i E i
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Equations involving Logarithmic Expressions
1 log2(log2(log2 x )) = 12 log(x +1)(3x + 1) = 2
3 logx 2 + logx 5 = 1
24 2log5(x − 2)− log5 x =
log5(x + 1)
51
2
log3(x 2+6) = log3 4
−log3 x
6 log4(x log4 x ) = 4
7 (log3 x )2 − log3 x 2 = log2 8
8 32x −1 = 4x +2
9 e 2x − e
x
= 610 ln
√ x =
√ ln x
11 log3 x + log9 x + log27 x = 5.5
12 3x + 3x = 32x
13 32−x − 108 · 2x −1 = 0
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Prepare for Quiz 1 next
meeting!
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