Exponential and Logarithmic Functions.pdf

8/18/2019 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 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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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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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 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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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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Euler’s Number



e ≈ 2.718282

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Euler’s Number



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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Definition

The natural exponential function is the exponential function

with base e :f (x ) = e x .

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Equations involving Exponential Expressions

Exercises

Solve for the solution set of the following equations:

1 53x = 57x −2

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Exercises


1 53x = 57x −2

2 4t 2

= 23t +2

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p pExponential FunctionsLogarithmic Functions


Exercises


1 53x = 57x −2

2 4t 2

= 23t +2

3 45−9x = 1

8x −2

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Exponential FunctionsLogarithmic Functions


Exercises


1 53x = 57x −2

2 4t 2

= 23t +2

3 45−9x = 1

8x −24

9x

+ 2(3x

) − 3 = 0

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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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Logarithms


Exercises

Let a ∈ R, a > 0 and a = 1.1 log4 16 =

2 log51

125 =

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Logarithms


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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Logarithms


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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Logarithms


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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Relationship of Exponential and Logarithmic Functions

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Let b ∈R

such that b > 0 and b = 1.

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pLogarithmic 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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pLogarithmic 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 .


x = b y

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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 .


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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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 .


x = b y


logb x = y

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Properties of ExponentsExponential FunctionsL ith i F ti

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Let b ∈R

such that b >


f (x ) = b x .


x = b y


logb x = y

Thus,

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Let b ∈R

such that b >


f (x ) = b x .


x = b y


logb x = y

Thus,f −1(x ) = logb x

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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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f (x ) = logb x


Notes:

1 dom f = (0, +∞)2 ran f = R

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f (x ) = logb x


Notes:

1 dom f = (0, +∞)2 ran f = R

3 logb (b x ) = x for all x ∈ R

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g u



f (x ) = logb x


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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g



f (x ) = logb x


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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f (x ) = logb x


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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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


2 Evaluate log6 4 + log6 9 − log6 5 − log6 3

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Exercises


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



7

x 2 + xy


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



7

x 2 + xy


logarithms.

4 Express 1

2 logb m +

3



5 Given loga 2 = 0.3 and loga 3 = 0.48. Find loga 72.

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Exercises



7

x 2 + xy


logarithms.

4 Express 1

2 logb m +

3



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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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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