Transcript 3.2 Logarithmic Functions

3
Exponential and
Logarithmic Functions
Contents
3.1 Rational Indices
3.2 Logarithmic Functions
3.3 Graphs of Exponential and Logarithmic
Functions
3.4 Applications of Logarithms
3
Exponential and Logarithmic Functions
3.1 Rational Indices
A. Radicals
For a positive integer n, if x n  y , then x is and n th root of y, denoted by
the
. radical n y
However, if n is even and y > 0, then x   n y or  n y is the solution of the
equation xn = y.
But in this chapter, we shall only consider the positive value of x.
If x n  y , then x 
n
y.
Remarks:
Content
1.
For n = 2, we call x the square root of y.
For n = 3, we call x the cube root of y.
2.
2
y
is usually written as
y.
P. 2
3
Exponential and Logarithmic Functions
3.1 Rational Indices
B. Rational Indices
The laws of indices are also true for rational indices.
For y  0 , we define the rational indices as follows:
1
yn 
n
y
m
y n  (n y )
m

n
y
m
where m, n are integers and n > 0.
Content
P. 3
3
Exponential and Logarithmic Functions
3.1 Rational Indices
C. Using the Laws of Indices to Solve Equations
p
For equation x  b where b is a non-zero constant, p and q are integers with
q
q0
q
, we take the power of p on both sides,
p
q
q
p
(x )
p
x
q

q
b
q
p
p
q
b
p
q
Content

xb
p
P. 4
3
Exponential and Logarithmic Functions
3.2 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  log a y .
If y  a x , then log a y  x, where a > 0 and a  1 .
Content
P. 5
3
Exponential and Logarithmic Functions
3.2 Logarithmic Functions
Notes :
Content
1.
If y  0 , then log a y is undefined.
2.
When a = 10 (thus base 10), we write log x for log10 x. This is called
the common logarithm.
3.
By the definition of logarithm and the laws of indices, we can obtain the
following results directly.
(a) 
1  10 ,

log 1  0
(b) 
10  10 ,

log 10  1
(c)  100  10 2 ,

log 100  2
0 . 1  10 ,
1

log 0 . 1   1
(e)  0 . 01  10  2 ,

log 0 . 01   2
(d) 
0
1
P. 6
3
Exponential and Logarithmic Functions
3.2 Logarithmic Functions
B. Basic Properties of Logarithmic Functions
The function f ( x )  log x , for x > 0 is called a logarithmic function.
Properties of Logarithmic functions:
For M, N > 0,
1. log( MN )  log M  log N
2. log
Content
M
 log M  log N
N
n
3. log M  n log M
P. 7
3
Exponential and Logarithmic Functions
3.2 Logarithmic Functions
C. Using Logarithms to Solve Equations
(a) Logarithmic Equations
Logarithmic equations are the equations containing the logarithm
of one or more variables.
For example, log x = 2 is a logarithmic equation.
In order to solve these kinds of equations, we need
to use the definition and the properties of logarithm.
Content
For example, if log x = 2, then
x  10
2
 100
P. 8
3
Exponential and Logarithmic Functions
3.2 Logarithmic Functions
(b) Exponential Equations
Exponential equations are the equations in the form ax = b, where a
and b are non-zero constants and a  1 .
For such equations, we take logarithm on both sides and reduce the
exponential equation to a linear equation, that is,
log a  log b
x
x log a  log b
Content
x
log b
log a
P. 9
3
Exponential and Logarithmic Functions
3.2 Logarithmic Functions
D. Other Types of Logarithmic Functions
For bases other than 10, such as the function f ( x )  log a x for x  0 , a  0
and a  1 , they are also called logarithmic functions.
The logarithmic functions with different bases still have the following
properties:
1.
log a a  1
2. log a 1  0
3.
log a ( MN )  log a M  log a N
4. log a
5.
log a M
Content
n
M
N
 log a M  log a N
 n log a M
P. 10
3
Exponential and Logarithmic Functions
3.3 Graphs of Exponential and Logarithmic Functions
A. Graphs of Exponential Functions
For a > 0 and a  1 , a function y = ax is called exponential function,
where a is the base and x is the exponent.
x
x
The following diagram shows the graph of y  2 and y  2 for –3  x  3.
Content
Fig. 3.2
P. 11
3
Exponential and Logarithmic Functions
3.3 Graphs of Exponential and Logarithmic Functions
Properties of the graph of exponential function:
Fig. 3.2
Content
1.
y = ax and y = a–x are reflectionally symmetric about the y-axis.
2.
The graph does not cut the x-axis (that is y > 0 for all values of x).
3.
The y-intercept is 1.
4.
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.
P. 12
3
Exponential and Logarithmic Functions
3.3 Graphs of Exponential and Logarithmic Functions
B. Graphs of Logarithmic Functions
Fig. 3.5 shows the graph of y = log x.
Content
Fig. 3.5
P. 13
3
Exponential and Logarithmic Functions
3.3 Graphs of Exponential and Logarithmic Functions
The function f (x) = 10x is called the inverse function of the common logarithmic
function f (x) = log x.
Properties of the graph of logarithmic
function:
Content
1.
The function is undefined for x  0 .
2.
For the graph of y = log x,
(a)
x-intercept is 1;
(b)
it does not have y-intercept;
(c)
y increases as x increases.
Fig. 3.5
P. 14
3
Exponential and Logarithmic Functions
3.4 Applications of Logarithms
A. Transforming Data from Exponential Form to Linear Form
We can actually transform data from exponential form to linear form.
Suppose y = kxn, where k > 0 and n  0.
Taking logarithm on both sides, we have
log y  log( kx )
n
log y  log k  n log x
Y  a  bX ,
Content
which is a linear function with
Y = log y, X = log x, a = log k and b = n.
P. 15
3
Exponential and Logarithmic Functions
3.4 Applications of Logarithms
B. Applications of Logarithms in Real-life Problems
1.
Loudness of Sound
Decibel (dB) is the unit for measuring the loudness L of sound,
which is defined as
L  10 log
I
,
I0
where I is the intensity of sound and I0 is the threshold of hearing
for a normal person.
Content
Notes:
I0 is the minimum audible sound intensity which is about 1012 W/m2.
P. 16
3
Exponential and Logarithmic Functions
3.4 Applications of Logarithms
2.
Richter Scale
The Richter scale (R) is a scale for measuring the magnitude of an
earthquake.
It is calculated from the energy E released from an earthquake
and is given by the following formula,
log E = 4.8 + 1.5R
where E is measured in joules (J).
Content
P. 17