Transcript Book 4 Chapter 5 Expo and Log Functions

5
Exponential and
Logarithmic Functions
Case Study
5.1
Rational Indices
5.2
Logarithmic Functions
5.3
Using Logarithms to Solve Equations
5.4
Graphs of Exponential and Logarithmic Functions
5.5
Applications of Logarithms
Chapter Summary
Case Study
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?
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.
P. 2
5.1 Rational Indices
A. Radicals
We learnt that if x2  y, for y  0, then x is a square root of y.
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 n y
to denote an nth root of y.
2
y is usually written as
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: n y ; Negative nth root:  n y

If n is an even number and y  0,
then n y is not a real number.

If n is an odd number,
then n y  0 (for  0) and
n
y  0 (for y  0).
Examples:
3
5
P. 3
64  4
 243  3
y.
5.1 Rational Indices
B. Rational Indices
In junior forms, we learnt the laws of indices for integral indices.
Recall that (am)n  amn, where m and n are integers.
Try to find the meaning of
1
( y n )n
1
yn
and
m
yn:
1
n
n
y

y
1
Take nth root on both sides, we have y n  n y.
Since
Then consider
Hence we have
m
yn

1
m
n
y
and
m
yn
y
m
1
n
1
 ( y n )m
1
 ( ym )n
 (n y ) m
 n ym
m
yn
 (n y ) m  n y m .
P. 4
5.1 Rational Indices
B. Rational Indices
Therefore, for y  0, we define rational indices as follows:
1.
1
yn
n y
2.
m
yn
 (n y ) m  n y m
where m, n are integers and n  0.
Remarks:

In the above definition, y is required to be positive.
However, the definition is still valid for y  0 under the following
situation:
m
The fraction
is in its simplest form and n is odd.
n
For example:
2
(8) 3
 (3  8) 2  (2) 2  4
P. 5
5.1 Rational Indices
B. Rational Indices
Example 5.1T
Simplify (3 b 2 ) 6, where b  0 and express the answer with
a positive index.
Solution:
(3 b 2 ) 6  (b

2
3 ) 6
2
 ( 6 )
b 3
 b4
First express the radical with
a rational index. Then use
(bm)n  bmn to simplify the
expression.
P. 6
5.1 Rational Indices
B. Rational Indices
Example 5.2T
1
 1
Evaluate  2 
 4

2
.
Solution:
 1
2 
 4

1
2
9
 
4

1
2
 1
2  
 3  2
 
2
1
3
 
2
2

3
Change the mixed number
into an improper fraction
first.
P. 7
5.1 Rational Indices
B. Rational Indices
Example 5.3T
9 x  3  3x  1
Simplify
.
x2
6  27
Solution:
9 x  3  3 x  1 3 2 ( x  3)  3 x  1

x2
6  27
2  3  33( x  2)
 1  32( x  3)  ( x  1)  1  3( x  2)
2
Change the numbers to the
same base before applying
the laws of indices.
1
  3 2
2
1 1
 
2 9
1

18
P. 8
5.1 Rational Indices
n
C. Using a Calculator to Find  y m
The following is the key-in sequence of finding 5 322 :
5
32
2
Since 5 322  32 5 ,
32
we can also press the following keys in sequence:
2
5
Remarks:
If n is an even number and ym  0, then ym has two nth real roots,
i.e.,  n y m . However, the calculator will only display
n
ym .
P. 9
5.1 Rational Indices
D. Using the Law of Indices to Solve Equations
p
q
For the equation x  b, where b is a non-zero constant,
p and q are integers with q  0, we can take the power of
q
on both sides and solve the equation:
p
p
q
3
x b
p q
q p
q
p
2
x3
p q

q p
q
p
2 3
(x3 )2
(x )  b
x
x2  4
b
xb
q
p
4
3
(22 ) 2

x 8
P. 10
5.1 Rational Indices
D. Using the Law of Indices to Solve Equations
Example 5.4T
If (2 x  1)

1
3
 1  3 , find x.
Solution:
(2 x  1)

1
3
1  3
(2 x  1)

1
3
1

[(2 x  1) 3 ]3
Change the equation into the
2
p
q
form x  b first.
 23
2x 1 
1
8
x
7
16
P. 11
5.1 Rational Indices
D. Using the Law of Indices to Solve Equations
Example 5.5T
Solve 22x  1  22x  8.
Solution:
22x  1  22x  8
2(22x)  22x  8
22x(2  1)  8
22x  8
22x  23
2x  3
3
x 
2
Take out the common factor
22x first.
P. 12
5.2 Logarithmic Functions
A. Introduction to Common Logarithm
If a number y can be expressed in the form ax, where a  0 and
a  1, then x is called the logarithm of the number y to the base a.
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.
If y  10n, then log y  n.
∴ log 10n  n for any real number n.
In the calculator, the button
log also stands for the
common logarithm.
P. 13
5.2 Logarithmic Functions
A. Introduction to Common Logarithm
By the definition of logarithm and the laws of indices, we can
obtain the following results directly:
1
1
 ∵
 102
∴ log
 2
100
100
1
1
 ∵
 101
∴ log  1
10
10
 ∵ 1  100
∴ log 1  0
 ∵ 10  101 ∴ log 10  1
 ∵ 100  102
∴ log 100  2
 ∵ 1000  103
∴ log 1000  3
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  101.2  0.0631 (cor. to 3 sig. fig.)
P. 14
5.2 Logarithmic Functions
B. Basic Properties of Common Logarithm
The function f (x)  log x, for x  0 is called a logarithmic function.
There are 3 important properties of logarithmic functions:
For M, N  0,
1. log (MN)  log M  log N
M
2. log
 log M  log N
N
3. log M n  n log M
In general,
1. log M  log N  log (MN);
log M
M
 log
2.
;
log N
N
3. (log M)n  log M n.
Let M  10a and N  10b. Then log M  a and log N  b.
 Consider MN  10a  10b  10a  b
Take common logarithm on both sides.
∴ log (MN)  a  b  log M  log N
M 10a
 Consider
 b  10a  b
 Consider M n  (10a)n
N
10
 10na
M
∴ log M n  na  n log M
∴ log
 a  b  log M  log N
N
P. 15
5.2 Logarithmic Functions
B. Basic Properties of Common Logarithm
Example 5.6T
Evaluate the following expressions.
3
2  log 25
(a)
log 5  log 80
(b)
2
log 8
Solution:
(a)
3
3
1
log 5  log 80  log 5  log 80
2
2
2
1
 (3 log 5  log 80)
2
Since 3 log 5  log 80  log 53  log 80
 log (53  80)
 log 10 000
4
3
∴
log 5  log 80  2
2
P. 16
5.2 Logarithmic Functions
B. Basic Properties of Common Logarithm
Example 5.6T
Evaluate the following expressions.
3
2  log 25
(a)
log 5  log 80
(b)
2
log 8
Solution:
2  log 25 log 100  log 25
(b)

log 8
log 8
 100 
log 

 25 

log 8
log 4

log 8

log 2 2
3
22
log
2 log 2

3
log 2
2
4

3
P. 17
5.2 Logarithmic Functions
B. Basic Properties of Common Logarithm
Example 5.7T
4 log x 5
Simplify
, where x  0.
3
3
log x  log x
Solution:
4(5) log x
4 log x 5

log x 3  log 3 x 3 log x  1 log x
3
20 log x

10
log x
3
6
P. 18
5.2 Logarithmic Functions
B. Basic Properties of Common Logarithm
Example 5.8T
If log 3  a and log 5  b, express the following in terms of a and b.
(a) log 225
(b) log 18
Solution:
(a)
log 225  log (32  52)
(b)
 log 32  log 52
 2 log 3  2 log 5
 2a  2b
log 18  log (2  32)
 log 2  log 32
10
 log  2 log 3
5
 log 10  log 5  2 log 3
 1  b  2a
P. 19
5.2 Logarithmic Functions
C. Other Types of Logarithmic Functions
We have learnt the logarithmic function with base 10 (i.e. common
logarithm).
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  1
2. loga 1  0
3. loga (MN)  loga M  loga N
M
4. loga
 loga M  loga N
N
5. loga M n  n loga M
P. 20
5.2 Logarithmic Functions
C. Other Types of Logarithmic Functions
Example 5.9T
1
Evaluate log 2 24  log 2 6 .
2
Solution:
1
log 2 24  log 2 6  log 2 24  log 2 6
2
24
 log 2
6
 log 2 4
 log 2 2
1
P. 21
5.2 Logarithmic Functions
D. Change of Base Formula
A calculator can only be used to find the values of common
logarithm.
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 Formula
For any positive numbers a and M with a  1, we have
log M
.
log a M 
log a
Let y  loga M, then we have a y  M.
log a y  log M
y log a  log M
log M
y
log a
Take common logarithm on both sides.
P. 22
5.2 Logarithmic Functions
D. Change of Base Formula
In fact, besides common logarithm, the change of base formula
can also be applied for logarithms with bases other than 10.
Change of Base Formula
For any positive numbers a, b and M with a, b  1, we have
log b M
.
log a M 
log b a
P. 23
5.2 Logarithmic Functions
D. Change of Base Formula
Example 5.10T
Solve the equation logx  1 8  25.
(Give the answer correct to 3 significant figures.)
Solution:
logx  1 8  25
log 8
 25
log ( x  1)
log 8
log (x  1) 
25
 0.03612
x  1  1.08673
x  0.0867 (cor. to 3 sig. fig.)
P. 24
5.2 Logarithmic Functions
D. Change of Base Formula
Example 5.11T
Show that log8 x 
2
log4 x, where x  0.
3
Solution:
log8 x 

log 4 x
log 4 8
log 4 x
log 4
3
42
log 4 x
3
log 4 4
2
2
 log 4 x
3

P. 25
5.3 Using Logarithms to Solve Equations
A. Logarithmic Equations
Logarithmic equations are equations containing the logarithm
of one or more variables.
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. 26
5.3 Using Logarithms to Solve Equations
A. Logarithmic Equations
Example 5.12T
Solve the equation log3 (x  7)  log3 (x  1)  2.
Solution:
log3 (x  7)  log3 (x  1)  2
 x 7
log 3 
2
 x 1 
x7
 32
x 1
x  7  9x  9
x 2
P. 27
5.3 Using Logarithms to Solve Equations
A. Logarithmic Equations
Example 5.13T
Solve the equation log x2  log 4x  log (x  1).
Solution:
log x2  log 4x  log (x  1)
 log 4x(x  1)
 log (4x2  4x)
x2  4x2  4x
3x2  4x  0
x(3x  4)  0
4
x
or 0 (rejected)
3
When x  0, log x2  log 0,
log 4x  log 0 and
log (x  1)  log (1), which
are undefined. So we have to
reject the solution of x  0.
P. 28
5.3 Using Logarithms to Solve Equations
B. Exponential Equations
Exponential equations are equations in the form ax  b, where
a and b are non-zero constants and a  1.
To solve ax  b:
ax  b
log ax  log b
x log a  log b
Take common logarithm on both sides.
The equation is reduced to linear form.
log b
∴ x 
log a
P. 29
5.3 Using Logarithms to Solve Equations
B. Exponential Equations
Example 5.14T
Solve the equation 5x  3  8x  1.
(Give the answer correct to 2 decimal places.)
Solution:
5x  3  8x  1
log 5x  3  log 8x  1
(x  3) log 5  (x  1) log 8
x(log 5  log 8)  3 log 5  log 8
 3 log 5  log 8
log 5  log 8
 14.70 (cor. to 2 d. p.)
x
P. 30
5.4 Graphs of Exponential and
Logarithmic Functions
A. Graphs of Exponential Functions
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.
Consider the exponential function y  2x.
x –1
y 0.5
0
1
1
2
2
4
3
8
 1
y 
2
x
y2
x
4 5
16 32
The domain of the function is all real
numbers.
y  0 for all real values of x.
x
1
Also plot the function y    :
2
x
y
–5 –4 –3 –2 –1
32 16 8 4 2
0
1
1
0.5
The graphs are reflectionally symmetric about the y-axis.
y-intercept  1
P. 31
5.4 Graphs of Exponential and
Logarithmic Functions
A. Graphs of Exponential Functions
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.
x
1


4. The graphs of y  ax and y    are reflectionally
a
symmetric about the y-axis.
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. 32
5.4 Graphs of Exponential and
Logarithmic Functions
A. Graphs of Exponential Functions
Consider the graphs of y  2x and y  3x.
x
y
–1
0.5
0
1
1
2
2
4
3
8
4
16
x
y
–1
0.33
0
1
1
3
2
9
3
27
4
81
y3
x
The graph y  3x increases more rapidly.
x
1
Consider the graphs of y    and
2
x
1
y    . Which graph decreases more
 3
rapidly?
P. 33
y2
x
5.4 Graphs of Exponential and
Logarithmic Functions
A. Graphs of Exponential Functions
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
rapidly as x increases;
(ii) If 1  b  a, then the graph of y  ax decreases more
rapidly as x increases.
P. 34
5.4 Graphs of Exponential and
Logarithmic Functions
B. Graphs of Logarithmic Functions
Consider the graphs of y  log2 x and y  log1 x.
2
x 0.5
y –1
1
0
2
1
4
2
8
3
16
4
x 0.5
y 1
1
0
2 4 8 16
–1 –2 –3 –4
x-intercept  1
The domain of the function is all positive real numbers.
The graphs are reflectionally symmetric about the x-axis.
P. 35
5.4 Graphs of Exponential and
Logarithmic Functions
B. Graphs of Logarithmic Functions
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. 36
5.4 Graphs of Exponential and
Logarithmic Functions
B. Graphs of Logarithmic Functions
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.
 by the algebraic method.
e.g., when x  7, y  log2 7
log 7
 2.8

log 2
If a value of y is given:


Graphical method
e.g., when y  1.6, x  3.0.
Algebraic method
e.g., when y  1.6, 1.6  log2 x
21.6  x
x  3.0
Given a value of y, then x  2y.
dependent variable
independent variable
inverse function
∴ f (x)  2x
f (x)  log2 x
P. 37
5.4 Graphs of Exponential and
Logarithmic Functions
C. Relationship between the Graphs of
Exponential and Logarithmic Functions
Consider the graphs of y  2x and y  log2 x.
y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.
P. 38
5.4 Graphs of Exponential and
Logarithmic Functions
D. Transformations on the Graphs of
Exponential and Logarithmic Functions
In Chapter 4, we learnt about the transformations of the graphs
of functions.
We can also transform logarithmic functions and exponential functions.
For example:
Let f (x)  log2 x and g(x)  log2 x  2.
y  log x  2
∴ y  f (x) is translated 2 units
upwards to become y  g(x).
2
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
P. 39
5.4 Graphs of Exponential and
Logarithmic Functions
D. Transformations on the Graphs of
Exponential and Logarithmic Functions
Example 5.15T
The following figure shows the graph of y  log2 x. Use the graph
to sketch the graphs of the following functions:
1
2
(a) y  log2 x
(b) y  log2
4
x
Solution:
1
(a) Since y  log2 x
4
1
 log2 x  log2
4
 log2 x – 2
∴
y  log2 1 x
4
1
the graph of y  log2 x is obtained by translating the
4
graph of y  log2 x two units downwards.
P. 40
5.4 Graphs of Exponential and
Logarithmic Functions
D. Transformations on the Graphs of
Exponential and Logarithmic Functions
Example 5.15T
The following figure shows the graph of y  log2 x. Use the graph
to sketch the graphs of the following functions:
1
2
(a) y  log2 x
(b) y  log2
4
x
Solution:
2
x
 log2 2 – log2 x
(b) Since y  log2
 1 – log2 x
i.e., y  –log2 x  1.
∴
y  log2 2
x
2
is obtained by reflecting the graph of
x
y  log2 x about the x-axis, then translating one unit upwards.
The graph of y  log2
P. 41
5.4 Graphs of Exponential and
Logarithmic Functions
D. Transformations on the Graphs of
Exponential and Logarithmic Functions
Example 5.16T
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  2x
Solution:
(a) Let f (x)  2x,
g(x)  2x  2.
y  2x + 2
∵ g(x)  f (x  2)
∴
the graph of y  2x  2 is obtained by translating the graph
of y  2x two units to the left.
P. 42
5.4 Graphs of Exponential and
Logarithmic Functions
D. Transformations on the Graphs of
Exponential and Logarithmic Functions
Example 5.16T
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  2x
Solution:
y  2x
(b) Let f (x)  2x,
h(x)  2x.
∵ h(x)  f (–x)
∴
the graph of y  2x is obtained by reflecting the graph
of y  2x about the y-axis.
P. 43
5.5 Applications of Logarithms
(a) Loudness of Sound
Decibel (dB): unit for measuring the loudness L of sound:
L  10 log
I
I0
where I is the intensity of sound and I0 ( 1012 W/m2) is the
threshold of hearing (minimum audible sound intensity) for
a normal person.
For example:
Given that I  103 W/m2.
103
∴ Loudness of sound  10 log 12 dB
10
 10 log 109 dB
 10(9) dB
 90 dB
W/m2 is the unit of the sound
intensity used in Physics,
which represents ‘watt per
square metre’.
P. 44
5.5 Applications of Logarithms
Sound intensity of 1 W/m2 is large enough to cause damage
to our audition (hearing):
1
Loudness of sound  10 log 12 dB
10
 10 log 1012 dB
 10(12) dB
 120 dB
which is about the loudness of airplane’s engine.
Loudness
20 dB
30 dB
40 dB
50 dB
60 dB
80 dB
100 dB
120 dB
Example
Camera shutter
A silent park
A silent class
An office with typical sound
A conversation between two people one metre apart
MTR platform
Motor car’s horn
Plane’s engine
P. 45
5.5 Applications of Logarithms
Example 5.17T
If 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.
I 80

 I 80
 I 80
8
80

10
log
log

8

10


 I
I0
I0
0



I
I
I
100  10 log 100
log 100  10
 100  1010


 I 0
I0
I0
I 80  108 I 0

10
I100  10 I 0
We can express each of the
sound intensities I80 and I100
∴ I80 : I100  108I0 : 1010I0
in terms of I0.
 1 : 100
P. 46
5.5 Applications of Logarithms
(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).
Remarks:
Examples of serious earthquakes on Earth:
 date: May 12, 2008
magnitude: 8.0
location: Sichuan province of China
 date: Dec 26, 2004
magnitude: 9.0
location: Indian Ocean
 date: May 22, 1960
magnitude: 9.5
location: Chile
The Richter scale was
developed by an American
scientist, Charles Richter.
P. 47
5.5 Applications of Logarithms
Example 5.18T
The 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.
You may notice that for
earthquakes with a difference
104.8  1.5(8.7 )
1.5(8.7 – 7.3)
in magnitude of 1.4 on the
 10
4.8  1.5( 7.3)
Richter scale, their energy
10
2.1
released is almost 125 times
 10
greater.
 126
∴ The energy released by the Tangshan earthquake was 126 times
that of Taiwan’s earthquake.
P. 48
Chapter Summary
5.1 Rational Indices
Laws of Indices
For a, b  0,
1.
am  an  am  n
2.
am  an  am  n
3.
(am)n  amn
4.
(ab)m  ambm
5.
am
a
   m
b
b
m
6.
1
an
n a
7.
m
an
 n am
P. 49
Chapter Summary
5.2 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 Logarithm
For any positive numbers a, b, M and N with a, b  1,
1. loga a  1
2. loga 1  0
3. loga (MN)  loga M  loga N
M
4. loga
 loga M  loga N
N
5. loga M n  n loga M
6.
loga M 
log b M
log b a
P. 50
Chapter Summary
5.3 Using Logarithms to Solve Equations
By using the properties of logarithm, we can solve the logarithmic
equations and exponential equations.
P. 51
Chapter Summary
5.4 Graphs of Exponential and Logarithmic
Functions
x
1.
2.
1
The graphs of y  a and y    are reflectionally symmetric
a
about the y-axis.
x
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 line
y  x.
P. 52
Chapter Summary
5.5 Applications of Logarithms
Daily-life applications of logarithms:
1. Measurement of loudness of sound in decibels (dB)
2. Measurement of magnitude of an earthquake on the Richter scale
P. 53