Exponential and Logarithmic Functions 5 5.1Rational Indices 5.2Logarithmic Functions Chapter Summary Case Study 5.4Graphs of Exponential and Logarithmic

Exponential and Logarithmic Functions5

5.1 Rational Indices

5.2 Logarithmic Functions

Chapter Summary

Case Study

5.4 Graphs of Exponential and Logarithmic Functions

5.3 Using Logarithms to Solve Equations

5.5 Applications of Logarithms

P. 2

Case Study

Although it is not easy to see bacteria with our naked eyes, bacteria do exist almost everywhere.Most bacteria reproduce by cell division, that is, one bacterium can divide and become 2, 2 become 4, and so on. However, each species reproduces itself at different rates due to different humidities and temperatures.

The growth rate of bacteria can be expressed by an exponential function. We can use it to find the number of bacteria.

How can we find the number of bacteria after a certain period of time?

P. 3

5.1 Rational Indices5.1 Rational Indices

We learnt that if x2 y, for y 0, then x is a square root of y.

A. RadicalsA. Radicals

If x3 y, then x is a cube root of y.

If x4 y, then x is a fourth root of y.

In general, for a positive integer n, if xn y, then x is an nth root of y and we use the radical to denote an nth root of y.

n y is usually written as .2 y y

Remarks: If n is an even number and y 0,

then y has one positive nth root and one negative nth root. Positive nth root: ; Negative nth root:n y n y

If n is an even number and y 0, then is not a real number. n y

If n is an odd number,then 0 (for 0) and 0 (for y 0).n y n y

Examples: 4643 32435

P. 4


In junior forms, we learnt the laws of indices for integral indices.

B. Rational IndicesB. Rational Indices

Recall that (am)n amn, where m and n are integers.

nnnn yy

11

)(Since y

Take nth root on both sides, we have .nn yy 1

Try to find the meaning of and :ny1

nm

y

Then consider and

mny

1

nm

y1

nm

y nm

y

mny )(1

nmy1

)(mn y)( n my

Hence we have .n mmnnm

yyy )(

P. 5


Therefore, for y 0, we define rational indices as follows:

B. Rational IndicesB. Rational Indices

1.

2.

where m, n are integers and n 0.

nn yy 1

n mmnnm

yyy )(

Remarks: In the above definition, y is required to be positive.

However, the definition is still valid for y 0 under the following situation:

The fraction is in its simplest form and n is odd.n

m

For example: 4)2()8()8( 22332

P. 6

5.1 Rational Indices5.1 Rational IndicesB. Rational IndicesB. Rational Indices

Example 5.1T

Solution:

4

)6(32

b

b

Simplify , where b 0 and express the answer with a positive index.

63 2 )( b

63 2 )( b 632

)( b

First express the radical with a rational index. Then use (bm)n bmn to simplify the expression.

P. 7


Example 5.2T

Solution:

Evaluate .21

4

12

Change the mixed number into an improper fraction first.

3

2

21

4

9

21

4

12

21

2

2

3

1

2

3

P. 8


Example 5.3T

Solution:

Change the numbers to the same base before applying the laws of indices.

18

1

Simplify .2

1 3

276

39

x

xx

232

1

9

1

2

1

)2 (3 1 )1 ( )3 (2321 xxx

2

1 3

276

39

x

xx

)2 (3

1 )3 (2

332

33

x

xx

P. 9

5.1 Rational Indices5.1 Rational IndicesC. Using a Calculator to Find C. Using a Calculator to Find y y mmnn

The following is the key-in sequence of finding : 5 232

5 32

Since , we can also press the following keys in sequence:52

5 2 3232

32 2 5 Remarks:If n is an even number and ym 0, then ym has two nth real roots,

i.e., . However, the calculator will only display .n my n my

P. 10

5.1 Rational Indices5.1 Rational IndicesD. Using the Law of Indices to Solve EquationsD. Using the Law of Indices to Solve Equations

For the equation b, where b is a non-zero constant, p and q are integers with q 0, we can take the power of

on both sides and solve the equation:

qp

x

p

q

pq

pq

pq

qp

pq

pq

qp

qp

bx

bx

bx

bx

)(

8)2()(

4

4

23

223

32

32

3 2

xx

x

x

P. 11


Example 5.4T

Solution:

If , find x.31)12( 31

x

31)12( 31

x

2)12( 31

x

3331

2])12[( x

8

112 x

16

7x

Change the equation into the

form first.bx qp

D. Using the Law of Indices to Solve EquationsD. Using the Law of Indices to Solve Equations

P. 12


Example 5.5T

Solution:

Solve 22x 1 22x 8.

Take out the common factor 22x first.

22x 1 22x 8 2(22x) 22x 8 22x(2 1) 8 22x 8 22x 23

2x 3

x 2

3

D. Using the Law of Indices to Solve EquationsD. Using the Law of Indices to Solve Equations

P. 13

If a number y can be expressed in the form ax, where a 0 anda 1, then x is called the logarithm of the number y to the base a.

5.2 Logarithmic Functions5.2 Logarithmic FunctionsA. Introduction to Common LogarithmA. Introduction to Common Logarithm

It is denoted by x loga y.

If y ax, then loga y x, where a 0 and a 1.

Notes: If y 0, then loga y is undefined.

Thus the domain of the logarithmic function loga y is the set of all positive real numbers of y.

When a 10 (base 10), we write log y for log10 y.This is called the common logarithm.

In the calculator, the button log also stands for the common logarithm.

If y 10n, then log y n.

∴ log 10n n for any real number n.

P. 14

By the definition of logarithm and the laws of indices, we can obtain the following results directly:

5.2 Logarithmic Functions5.2 Logarithmic FunctionsA. Introduction to Common LogarithmA. Introduction to Common Logarithm

∵ 1 100 log 1 ∴ 0 ∵ 10 101 log 10 ∴ 1

∵ 100 102 log 100 ∴ 2

∵ 1000 103 log 1000 ∴ 3

∵ 102 log ∴ 2100

1

100

1

∵ 101 log ∴ 110

1

10

1

Values other than powers of 10 can be found by using a calculator.

For example: log 34 1.5315

(cor. to 4 d. p.)

Given log x 1.2. ∴ x 101.2 0.0631 (cor. to 3 sig. fig.)

P. 15

The function f (x) log x, for x 0 is called a logarithmic function.

5.2 Logarithmic Functions5.2 Logarithmic FunctionsB. Basic Properties of Common LogarithmB. Basic Properties of Common Logarithm

There are 3 important properties of logarithmic functions:

For M, N 0,1. log (MN) log M log N

2. log log M log N

3. log M n n log MN

M

Let M 10a and N 10b.

Then log M a and log N b.

Consider MN 10a 10b 10a b

∴ log (MN) a b

Consider N

M b

a

10

10 10a b

log M log N ∴ log a b N

M

Consider M n (10a)n

log M log N Take common logarithm on both sides.

∴ log M n na n log M 10na

In general,1. log M log N log (MN);

2. ;

3. (log M)n log M n.

N

M

N

M log

log

log

P. 16


Example 5.6T

Solution:

Evaluate the following expressions.

(a) log 5 log (b)2

380

8 log

25 log2

80log5log2

3 (a) 80log2

15log

2

3

)80log5log3(2

1

Since 3 log 5 log 80 log 53 log 80

log 10 000 log (53 80)

4

280log5log2

3 ∴

P. 17


Example 5.6T

Solution:

Evaluate the following expressions.

(a) log 5 log (b)2

380

8 log

25 log2

8log25

100 log

8log

4 log

2 log 23

2 log 2

(b)8log

25log2 8log

25 log100 log 23

2

2 log

2 log

3

4

P. 18

Example 5.7T

Solution:

Simplify , where x 0.33

5

log log

log 4

xx

x


33

5

log log

log 4

xx

x

xx

x

log 31

log 3

log )5(4

6

x

x

log 3

10 log 20

P. 19


Solution:

If log 3 a and log 5 b, express the following in terms of a and b.(a) log 225 (b) log 18

(a) log 225 log (32 52) log 32 log 52

2 log 3 2 log 5 2a 2b

(b) log 18 log (2 32) log 2 log 32

log 2 log 35

10

log 10 log 5 2 log 3 1 b 2a

Example 5.8T

P. 20

We have learnt the logarithmic function with base 10 (i.e. common logarithm).

5.2 Logarithmic Functions5.2 Logarithmic FunctionsC. Other Types of Logarithmic FunctionsC. Other Types of Logarithmic Functions

For logarithmic functions with bases other than 10, such as the function f (x) loga x for x 0, a 0 and a 1, they still have the following properties:

For M, N, a 0 and a 1,1. loga a 12. loga 1 03. loga (MN) loga M loga N

4. loga loga M loga N

5. loga M n n loga M

N

M

P. 21

Example 5.9T

Solution:

5.2 Logarithmic Functions5.2 Logarithmic Functions

Evaluate .6log2

124log 22

1

6log2

124log 22 6log24log 22

6

24log2

4log2

2log2

C. Other Types of Logarithmic FunctionsC. Other Types of Logarithmic Functions

P. 22

A calculator can only be used to find the values of common logarithm.

5.2 Logarithmic Functions5.2 Logarithmic FunctionsD. Change of Base FormulaD. Change of Base Formula

For logarithmic functions with bases other than 10, we need to use the change of base formula to transform the original logarithm into common logarithm:

Change of Base FormulaFor any positive numbers a and M with a 1, we have

.a

MMa log

loglog

Let y loga M, then we have a y M. log a y log M

y log a log M

a

My

log

log

Take common logarithm on both sides.

P. 23

In fact, besides common logarithm, the change of base formula can also be applied for logarithms with bases other than 10.


Change of Base FormulaFor any positive numbers a, b and M with a, b 1, we have

.a

MM

b

ba log

loglog

P. 24

Example 5.10TSolve the equation logx 1 8 25. (Give the answer correct to 3 significant figures.)


Solution:

logx 1 8 25

25)1( log

8 log

x

log (x 1) 25

8 log

0.03612 x 1 1.08673 x 0.0867 (cor. to 3 sig.

fig.)

P. 25

Example 5.11T


Solution:

Show that log8 x log4 x, where x 0. 3

2

8log

log

4

4 xlog8 x

23

4

4

4log

log x

4log23log

4

4 x

x4log3

2

P. 26

Logarithmic equations are equations containing the logarithm of one or more variables.

5.3 Using Logarithms to Solve Equations5.3 Using Logarithms to Solve EquationsA. Logarithmic EquationsA. Logarithmic Equations

For example: log x 2 log5 (x 2) 1

We need to use the definition and the properties of logarithm to solve logarithmic equations.

For example:If loga x 2, then x a2 .

P. 27


Solution:

Solve the equation log3 (x 7) log3 (x 1) 2.

log3 (x 7) log3 (x 1) 2

Example 5.12T

21

7 log3

x

x

231

7

x

x

x 7 9x 9 x 2

P. 28


Solution:

Solve the equation log x2 log 4x log (x 1).

log x2 log 4x log (x 1)

Example 5.13T

log 4x(x 1) log (4x2 4x) x2 4x2 4x 3x2 4x 0x(3x 4) 0 x or 0 (rejected)

3

4

When x 0, log x2 log 0, log 4x log 0 andlog (x 1) log (1), which are undefined. So we have to reject the solution of x 0.

P. 29

Exponential equations are equations in the form ax b, where a and b are non-zero constants and a 1.

5.3 Using Logarithms to Solve Equations5.3 Using Logarithms to Solve EquationsB. Exponential EquationsB. Exponential Equations

To solve ax b:

Take common logarithm on both sides.

ax b log ax log bx log a log b

The equation is reduced to linear form.

∴ x a

b

log

log

P. 30

Example 5.14TSolve the equation 5x 3 8x 1. (Give the answer correct to 2 decimal places.)

Solution:

5x 3 8x 1

5.3 Using Logarithms to Solve Equations5.3 Using Logarithms to Solve EquationsB. Exponential EquationsB. Exponential Equations

log 5x 3 log 8x 1

(x 3) log 5 (x 1) log 8 x(log 5 log 8) 3 log 5 log 8

x 8 log 5 log

8 log 5 log 3

14.70 (cor. to 2 d. p.)

P. 31

For a 0 and a 1, a function y ax is called an exponential function, where a is the base and x is the exponent.

A. Graphs of Exponential FunctionsA. Graphs of Exponential Functions

5.4 Graphs of Exponential and5.4 Graphs of Exponential and Logarithmic FunctionsLogarithmic Functions

y 2xConsider the exponential function y 2x.

x –1 0 1 2 3 4 5

y 0.5 1 2 4 8 16 32

Also plot the function y :x

2

1

The domain of the function is all real numbers.

y 0 for all real values of x.

The graphs are reflectionally symmetric about the y-axis.

x –5 –4 –3 –2 –1 0 1

y 32 16 8 4 2 1 0.5

y x

2

1

y-intercept 1

P. 32

Properties of the graphs of exponential functions:



1. The domain of exponential function is the set of all real numbers.

2. The graph does not cut the x-axis, that is, y 0 for all real values of x.

3. The y-intercept is 1.

4. The graphs of y ax and y are reflectionally

symmetric about the y-axis.

x

a

1

5. For the graph of y ax,(a) if a 1, then y increases as x increases.(b) if 0 a 1, then y decreases as x increases.

Notes:Property 4 can be proved algebraically. The graphs of y f (x) and

y g(x) are reflectionally symmetric about the y-axis if g(x) f (x).

P. 33

Consider the graphs of y 2x and y 3x.



y 2xx –1 0 1 2 3 4

y 0.5 1 2 4 8 16

x –1 0 1 2 3 4

y 0.33 1 3 9 27 81

The graph y 3x increases more rapidly.

y 3x

Consider the graphs of y and

y . Which graph decreases more

rapidly?

x

2

1

x

3

1

P. 34

Properties of the graphs of exponential functions:



For the graphs of y ax and y bx, where a, b, x 0,(i) If a b 1, then the graph of y ax increases more ra

pidly as x increases;(ii) If 1 b a, then the graph of y ax decreases more ra

pidly as x increases.

P. 35

B. Graphs of Logarithmic FunctionsB. Graphs of Logarithmic Functions


x 0.5 1 2 4 8 16

y –1 0 1 2 3 4

The domain of the function is all positive real numbers.

x 0.5 1 2 4 8 16

y 1 0 –1 –2 –3 –4

Consider the graphs of y log2 x and y log1 x. 2

The graphs are reflectionally symmetric about the x-axis.

x-intercept 1

P. 36

Properties of the graphs of logarithmic functions:


1. The domain of logarithmic function is the set of all positive real numbers, i.e., undefined for x 0.

2. The graph does not cut the y-axis, (that is, x 0 for all real values of y).

3. The x-intercept is 1.

4. The graphs of y loga x and y log1 x, where a 0 are

a

reflectionally symmetric about the x-axis.5. For the graph of y loga x,(a) if a 1, then y increases as x increases.(b) if 0 a 1, then y decreases as x increases.


P. 37

Consider the graph of y log2 x.



If a positive value of x is given, then the corresponding value of y can be found

by the graphical method.e.g., when x 7, y 2.8.

If a value of y is given:

by the algebraic method.e.g., when x 7, y log2

72 log

7 log 2.8

Graphical methode.g., when y 1.6, x 3.0.

Algebraic methode.g., when y 1.6, 1.6 log2 x 21.6

x x 3.0

Given a value of y, then x 2y.

independent variabledependent variable

∴ f (x) 2x f (x) log2 x

inverse function

P. 38

C. Relationship between the Graphs ofC. Relationship between the Graphs of Exponential and Logarithmic FunctionsExponential and Logarithmic Functions


Consider the graphs of y 2x and y log2 x.

Each of the functions f (x) 2x and y log2 x is the inverse function of each other.The graphs of y 2x and y log2 x are reflectional images of each other about the line y x.

y x

P. 39


In Chapter 4, we learnt about the transformations of the graphs of functions.

D. Transformations on the Graphs ofD. Transformations on the Graphs of Exponential and Logarithmic FunctionsExponential and Logarithmic Functions

We can also transform logarithmic functions and exponential functions.

For example:Let f (x) log2 x and g(x) log2 x 2.∴ y f (x) is translated 2 units

upwards to become y g(x).

However, we have to pay attention to the properties of logarithmic functions such as loga (MN) loga M loga N.

For example:If h(x) log2 4x, then it is the same as g(x):

log2 4x log2 4 log2 x 2 log2 x

y log2 x 2

P. 40


D. Transformations on the Graphs ofD. Transformations on the Graphs of Exponential and Logarithmic FunctionsExponential and Logarithmic FunctionsExample 5.15T

Solution:

The following figure shows the graph of y log2 x. Use the graph to sketch the graphs of the following functions:

(a) y log2 x (b) y log24

1

x

2

(a) Since y log2 x4

1

log2 x log2 4

1

log2 x – 2

y log2 x14

∴ the graph of y log2 x is obtained by translating the

graph of y log2 x two units downwards.4

1

P. 41



Solution:

The following figure shows the graph of y log2 x. Use the graph to sketch the graphs of the following functions:

(a) y log2 x (b) y log24

1

x

2

(b) Since y log

2x

2

log2 2 – log2 x

1 – log2 x

y log2 2x

∴ The graph of y log2 is obtained by reflecting the graph of

y log2 x about the x-axis, then translating one unit upwards.x

2

i.e., y –log2 x 1.

P. 42



Solution:

The following figure shows the graph of y 2x. Use the graph to sketch the graphs of the following functions:

(a) y 2x 2 (b) y 2x

(a) Let f (x) 2x,

∵ g(x) f (x 2)

g(x) 2x 2.

∴ the graph of y 2x 2 is obtained by translating the graph

of y 2x two units to the left.

y 2x + 2

P. 43



Solution:

The following figure shows the graph of y 2x. Use the graph to sketch the graphs of the following functions:

(a) y 2x 2 (b) y 2x

(b) Let f (x) 2x,

∵ h(x) f (–x)

h(x) 2x.

∴ the graph of y 2x is obtained by reflecting the graph

of y 2x about the y-axis.

y 2x

P. 44

(a) Loudness of Sound

5.5 Applications of Logarithms5.5 Applications of Logarithms

Decibel (dB): unit for measuring the loudness L of sound:

L 10 log0I

I

where I is the intensity of sound and I0 ( 1012 W/m2) is the threshold of hearing (minimum audible sound intensity) for a normal person.

W/m2 is the unit of the sound intensity used in Physics, which represents ‘watt per square metre’.

For example:Given that I 103 W/m2.

∴ Loudness of sound 10 log dB

12

3

10

10

10 log 109 dB 10(9) dB 90 dB

P. 45

Sound intensity of 1 W/m2 is large enough to cause damage to our audition (hearing):


Loudness of sound 10 log dB

1210

1

10 log 1012 dB 10(12) dB 120 dB

which is about the loudness of airplane’s engine.

Loudness Example

20 dB Camera shutter

30 dB A silent park

40 dB A silent class

50 dB An office with typical sound

60 dB A conversation between two people one metre apart

80 dB MTR platform

100 dB Motor car’s horn

120 dB Plane’s engine

P. 46


Example 5.17TIf one person makes noise of 80 dB and another makes noise of 100 dB, then what is the ratio of the sound intensities made by the two people?

Solution:

Let I80 and I100 be the sound intensities made by the two people respectively.

0

100

0

80

log 10100

log 1080

I

II

I

10log

8log

0

100

0

80

I

II

I

10

0

100

8

0

80

10

10

I

II

I

010

100

08

80

10

10

II

II

∴ I80 : I100 108I0 : 1010I0

We can express each of the sound intensities I80 and I100 in terms of I0. 1 : 100

P. 47

(b) Richter Scale


The Richter scale R is a scale used to measure the magnitude of an earthquake:

log E 4.8 1.5R

where E is the energy released from an earthquake, measured in joules (J).

The Richter scale was developed by an American scientist, Charles Richter.

Remarks:Examples of serious earthquakes on Earth:

date: Dec 26, 2004 magnitude: 9.0

location: Indian Ocean

date: May 12, 2008 magnitude: 8.0

location: Sichuan province of China

date: May 22, 1960 magnitude: 9.5

location: Chile

P. 48


Example 5.18TThe most serious earthquake occurred in China was the Tangshan earthquake, which occurred on July 28, 1976. It was recorded as having the magnitude of 8.7 on the Richter scale.Compare the energy released by that earthquake with Taiwan’s 9-21 earthquake with a magnitude of 7.3 on the Richter scale in 1999.

Solution:

Since log E 4.8 1.5R, E 104.8 1.5R.

)3.7(5.1 8.4

)7.8(5.1 8.4

10

10

101.5(8.7 – 7.3)

102.1

126

∴ The energy released by the Tangshan earthquake was 126 times that of Taiwan’s earthquake.

You may notice that for earthquakes with a difference in magnitude of 1.4 on the Richter scale, their energy released is almost 125 times greater.

P. 49

5.1 Rational Indices

Chapter Summary

Laws of IndicesFor a, b 0,

1. am an am n

2. am an am n

3. (am)n amn

4. (ab)m ambm

5.

6.

7.

m

mm

b

a

b

a

nn aa 1

n mnm

aa

P. 50

Chapter Summary5.2 Logarithmic Functions

If y ax, then loga y x, where a 0 and a 1.For base 10, we may write log y instead of log10 y.

Properties of LogarithmFor any positive numbers a, b, M and N with a, b 1,1. loga a 12. loga 1 03. loga (MN) loga M loga N

4. loga loga M loga N

5. loga M n n loga M

6. loga M

N

M

a

M

b

b

log

log

P. 51

Chapter Summary5.3 Using Logarithms to Solve Equations

By using the properties of logarithm, we can solve the logarithmic equations and exponential equations.

P. 52

Chapter Summary5.4 Graphs of Exponential and Logarithmic Functions

1. The graphs of y ax and y are reflectionally symmetric

about the y-axis.

x

a

1

2. The graphs of y loga x and y log1 x are reflectionally symmetric

a

about the x-axis.3. The graphs of y ax and y loga x are symmetric about the lin

e y x.

P. 53

Chapter Summary

Daily-life applications of logarithms:

5.5 Applications of Logarithms

1. Measurement of loudness of sound in decibels (dB)

2. Measurement of magnitude of an earthquake on the Richter scale

Documents

Exponential and Logarithmic Functions 5 5.1Rational Indices 5.2Logarithmic Functions Chapter Summary Case Study 5.4Graphs of Exponential and Logarithmic