Exponential and logarithmic functions - Cambridge

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C H A P T E R

5

E

Exponential and

logarithmic functions

Objectives

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To graph exponential and logarithmic functions.

To graph transformations of the graphs of exponential and logarithmic functions.

To introduce Euler¡¯s number.

To revise the index and logarithm laws.

To solve exponential and logarithmic equations.

SA

M

To find rules for the graphs of exponential and logarithmic functions.

To find inverses of exponential and logarithmic functions.

To apply exponential functions to physical occurrences of exponential growth and

decay.

5.1

Exponential functions

The function, f (x) = a x , where a ¡Ê R + \{1}, is called an exponential function (or index

function).

y

The graph of f (x) = a x is shown.

The features of the graph of the exponential

function with rule f (x) = a x are:

1

f (?1) =

a

(1, a)

f (0) = 1

f (1) = a

¨C1, 1

(0, 1)

a

The x-axis is a horizontal asymptote.

0

As x ¡ú ?¡Þ, f (x) ¡ú 0+ .

The maximal domain is R.

The range of the function is R + .

An exponential function is a one-to-one function.

174

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Chapter 5 ¡ª Exponential and logarithmic functions

175

Graphing transformations of the graph

of f (x) = ax

Translations

E

If the transformation, a translation with mapping (x, y) ¡ú (x + h, y + k), is applied to the

graph of y = a x , the image has equation y = a x?h + k. The
horizontal

 asymptote has

1

equation y = k. The images of the points with coordinates ?1,

, (0, 1) and (1, a) are

a






1

?1 + h, + k , (h, 1 + k) and (1 + h, a + k) respectively. The range of the image is

a

(k, ¡Þ).

Example 1

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Sketch the graph, and state the range, of y = 2x?1 + 2

Solution

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M

A translation of 1 unit in the positive direction of the x-axis and 2 units in the positive

y

direction of the y-axis is applied to the graph of y = 2x

The equation of the asymptote is y = 2

The mapping is (x, y) ¡ú (x + 1, y + 2)











5

1

¡ú 0,

?1,

2

2

(2, 4)

5

(0, 1) ¡ú (1, 3)

0,

(1, 3)

2

(1, 2) ¡ú (2, 4)

The range of the function is (2, ¡Þ).

2

0

x

Reflections

If the transformation, a reflection in the x-axis determined by the mapping (x, y) ¡ú (x, ?y),

x

is applied to the graph of y = a x , the image has equation y = ?a


. Thehorizontal asymptote

1

has equation y = 0. The images of the points with coordinates ?1, , (0, 1) and (1, a) are

a






1

?1, ? , (0, ?1) and (1, ?a) respectively. The range of the image is (?¡Þ, 0).

a

Example 2

Sketch the graph of y = ?3x

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176

Essential Mathematical Methods 3 & 4 CAS

Solution

A reflection in the x-axis is applied to the graph of y = 3x

The mapping is (x, y) ¡ú (x, ?y)








1

1

¡ú ?1, ?

3

3

(0, 1) ¡ú (0, ?1)

(1, 3) ¡ú (1, ?3)




0

(0, ¨C1)

x

(1, ¨C3)

E

?1,

1

¨C1, ¨C

3

y

If the transformation, a reflection in the y-axis determined by the mapping (x, y) ¡ú (?x, y),


x 




1

1

x

?x

is applied to the graph of y = a , the image has equation y = a

or y = x or y =

.

a

a

The

asymptote has equation


horizontal






 y = 0. The images of the points with coordinates

1

1

?1,

, (0, 1) and (1, a) are 1,

, (0, 1) and (?1, a) respectively. The range of the

a

a

image is (0, ¡Þ).

Example 3

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Sketch the graph of y = 6?x

Solution

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M

A reflection in the y-axis is applied to the graph of y = 6x

The mapping is (x, y) ¡ú (?x, y)








1

1

¡ú 1,

?1,

6

6

(0, 1) ¡ú (0, 1)

(1, 6) ¡ú (?1, 6)




y

(¨C1, 6)

(0, 1)

1,

1

6

0

Dilations

If the transformation, a dilation of factor k (k > 0) from the x-axis determined by the

mapping (x, y) ¡ú (x, ky), is applied to the graph of y = ax , the image has equation y = ka x .

The

asymptote has equation

y = 0. The images of the points with coordinates









horizontal

k

1

, (0, 1) and (1, a) are ?1,

, (0, k) and (1, ka) respectively. The range of the

?1,

a

a

image is (0, ¡Þ).

Example 4

Sketch the graph of each of the following:

a y = 3(5)x

b y = (0.2)(8)x

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Chapter 5 ¡ª Exponential and logarithmic functions

177

Solution

y

a A dilation of factor 3 from the x-axis is

applied to the graph of y = 5x

The mapping is (x, y) ¡ú (x, 3y)








3

1

¡ú ?1,

?1,

5

5

(0, 1) ¡ú (0, 3)

(1, 5) ¡ú (1, 15)

(1, 15)




¨C1,

3

5

(0, 3)

E

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x

0



1

from the x-axis

b A dilation of factor 0.2 or

5

is applied to the graph of y =
8x



1

The mapping is (x, y) ¡ú x, y

5











1

1

¡ú ?1,

?1,

8


 40

1

1

¨C1,

(0, 1) ¡ú 0,

40

5






8

(1, 8) ¡ú 1,

5

y

PL




0,

1,

8

5

1

5

x

0

SA

M

If the transformation, a dilation of factor k (k > 0) from the y-axis determined by the

x

mapping (x, y) ¡ú (kx, y), is applied to the graph of y = ax , the image has equation y = a k . The

horizontal

 asymptote has equation


y = 0.

 The images of the points with coordinates




1

1

, (0, 1) and (1, a) are ?k,

, (0, 1) and (k, a) respectively. The range of the

?1,

a

a

image is (0, ¡Þ).

Example 5

Sketch the graph of each of the following:

x

a y = 92

b y = 23x

Solution

y

a A dilation of factor 2 from the y-axis is applied

to the graph of y = 9x

The mapping is (x, y) ¡ú (2x, y)









1

1

¡ú ?2,

9

9

(0, 1) ¡ú (0, 1)

(1, 9) ¡ú (2, 9)

(2, 9)



?1,

¨C2,

1

9

(0, 1))

0

x

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178

Essential Mathematical Methods 3 & 4 CAS

1

from the y-axis is

3

applied to the graph of y = 2
x



1

x, y

The mapping is (x, y) ¡ú

3











1 1

1

¡ú ? ,

?1,

2

3 2

(0, 1) ¡ú (0, 1)






1

,2

(1, 2) ¡ú

3

y

b A dilation of factor

Example 6

1 1

¨C ,

3 2

(0, 1)

0

x

E

Combinations of transformations

1

,2

3

PL

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Sketch the graph, and state the range, of each of the following:

b y = 43x ? 1

c y = ?10x?1 ? 2

a y = 2?x + 3

Solution

SA

M

a The transformations, a reflection in the

y-axis and a translation of 3 units in the

positive direction of the y-axis,

are applied to the graph of y = 2x

The equation of the asymptote is y = 3

The mapping is (x, y) ¡ú (?x, y + 3)











7

1

¡ú 1,

?1,

2

2

(0, 1) ¡ú (0, 4)

(1, 2) ¡ú (?1, 5)

y

(0, 4) 1, 7

2

(¨C1, 5)

3

x

0

The range of the function is (3, ¡Þ).

1

from the y-axis followed by a translation

3

of 1 unit in the negative direction of the y-axis, are applied to the graph of y = 4x

The equation of the asymptote

y


is y = ?1



1

The mapping is (x, y) ¡ú

x, y ? 1

3

1

,3











3

1

3

1

¡ú ? ,?

?1,

4

3

4

(0, 1) ¡ú (0,

0)






1

(0, 0)

x

(1, 4) ¡ú

,3

0

3

b The transformations, a dilation of factor

The range of the function is (?1, ¡Þ).

1 3

¨C ,¨C

3 4

¨C1

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