MPC4C.01
Date: 2012/1/9
CHAPTER 5-1
RADICAL AND RATIONAL EXPONENTS
Wyan Chan
3) If c<0, then we will have no solution
Objectives:
-
Define and supply rational and irrational
exponents.
-
Simplify expressions containing radicals or
rational exponents.
Solution of
xn  c
If n is odd number…
nth Roots:
Then we can tell that there will be only exactly
can be written as
one solution for any c:
n
c
or
c
1
n
*Let c is a real number and n is a positive
integer.
Rational Exponents:
If n is even number…
1) If c>0, then we will know there will be one
t
k
1
t k
1
k t
c  (c )  (c )  k ct  ( k c )t
positive solution and one negative solution:
*Let c is a real number and also
t
is rational
k
number and k >0
Law of Exponents:
*Let c and d are nonnegative real numbers, and
also r and s = rational number:
These will be true:
2) If c=0, then we will have only one solution:
1) c r c s  c r  s
cr
 cr s
cs
3) (c r ) s  c rs
4) (cd )r  c r d r
2)
c
d
5) ( )r 
6) c  r 
1
cr
cr
dr
(d  0 )
(c  0 )
MPC4C.01
Date: 2012/1/9
*If c  1 d  1 ,
Wyan Chan
 y-intercept is 1

c  c if and only if r = s
 f(x) is decreasing

c d
 f(x) approaches the positive x-axis as x
r
r
s
r
if and only if c = d
approaches  .
Transformations:
CHAPTER 5-2
EXPONENTIAL FUNCTIONS
Translations:
Objectives:
Shift Horizontally:
-
Graph and identify transformations of
exponential functions.
-
f ( x  b)  a x  b
Use exponential functions to solve
If b>0, shifting left comparing to f(x)
application problems.
If b<0, shifting right comparing to f(x)
Exponential function with base a is a function
which follows f ( x)  a x .
Shift vertically:
f ( x)  c  a x  c
Graph of f ( x)  a x
If a>1:
If c>0, shifting up comparing to f(x)
If c<0, shifting down comparing to f(x)
Reflections:
Horizontal reflection:
f ( x)  a  x across y-axis
Vertical reflection:
 Graph is above x-axis
 Y-intercept is 1
 f(x) is increasing
 f(x) approaches the negative x-axis as x
approaches 
If 0<a<1
 f ( x)  a x across x-axis
Horizontal Stretches:
f (dx)  a dx
If d>0, compressing horizontally comparing to
f(x)
If d<0, stretched horizontally comparing to f(x)
Vertical Stretches:
h  f ( x)  ha x
If h>0, stretched vertically comparing to f(x)
If h<0, compressing vertically comparing to f(x)
 graph is above x-axis
MPC4C.01
Date: 2012/1/9
Application:
f ( x)  a
Wyan Chan
Exponential decay:
x
f ( x)  Pa x  P(1  r ) x
Growth, a>1
Decay, 0<a<1
Radioactive:
CHAPTER 5-3
x
f ( x)  P(0.5) h
APPLICATIONS OF EXPONENTIAL
FUNCTIONS
CHAPTER 5-4
Objectives:
COMMON AND NATURAL LOGARTHMIC
-
Create and use exponential models for a
FUNCTIONS
variety of exponential growth or decay
Objectives:
application problems.
-
For application problems, if p dollars is invested
at interest rate of r per time t, then A is the
Evaluate common and natural logarithms
with and without calculator.
-
Solve common and natural exponential and
amount after t period.
logarithmic equations by using an
A  P(1  r )
equivalent equation.
t
Tips*
Graph and identify transformations of
common and natural logarithmic functions.
Compounding
n
1
(1  ) n
n
Annually
1
2
Semiannually
2
2.25
Quarterly
4
2.4414
Monthly
12
2.6130
Daily
365
2.71457
Hourly
8760
2.718127
Every minute
525600
2.7182792
Every second
31536000
2.7182825
period
Common Logarithm:
Here is an example of the relationship between
exponential function and logarithmic function:
f ( x)  10 x
g ( x)  log x
It is its inverse.
If p dollars is invested at an annual interest rate
log v  u if and only if 10u  v
of r, compounded continuously, then A is the
amount after t years.
Natural Logarithms:
A  Pe rt
Here is an example of natural logarithms too
f ( x)  e x
Exponential growth:
f ( x)  Pa  P(1  r )
x
x
g ( x)  ln x
MPC4C.01
Date: 2012/1/9
Wyan Chan
If d>0, compressing horizontally comparing to
f(x)
If d<0, stretched horizontally comparing to f(x)
Vertical Stretches:
h  f ( x)  h ln( x)
Obviously, they are each other inverse function.
If h>0, stretched vertically comparing to f(x)
Therefore, ln v  u if and only if e  v
If h<0, compressing vertically comparing to f(x)
u
CHAPTER 5-5
GRAPH:
PROPERTIES AND LAWS OF LOGARITHMS
*When we graph logarithmic functions, we will
Objectives:
have to look at their inverse function, then
-
switch their point with x and y.
Use properties and laws of logarithms to
simplify and evaluate expressions.
Transformation:
Translations:
Basic Properties
Shift Horizontally:
-
log v and ln v are defined only when v >0
-
log1  0  ln1  0
-
log10k  k , for every real number k
If b>0, shifting left comparing to f(x)
-
ln e k  k , for every real number k
If b<0, shifting right comparing to f(x)
-
10log v  v  eln v  v for every v>0
Shift vertically:
For all v, w>0
f ( x  b)  ln( x  b)
f ( x)  c  ln x  c
-
log(vw)  log v  log w
ln(vw)  ln v  ln w
-
v
log( )  log v  log w
w
v
ln( )  ln v  ln w
w
If c>0, shifting up comparing to f(x)
If c<0, shifting down comparing to f(x)
Reflections:
Horizontal reflection:
f ( x)  ln( x) across y-axis
For all k and v>0,
log v k  k log v
Vertical reflection:
 f ( x)   ln( x) across x-axis
Horizontal Stretches:
f (dx)  ln(dx)
-
ln v k  k ln v
MPC4C.01
Date: 2012/1/9
Wyan Chan
CHAPTER 5-5.A
EXCURSION: LOGARITHMIC FUNCTIONS
GRAPH:
TO OTHER BASES
have to look at their inverse function, then
Objectives:
switch their point with x and y.
-
Evaluate logarithms to any base with and
without calculator
-
*Transformation is same as the previous ones.
Solve exponential and logarithmic equations
to any base by using an equivalent equation
-
Identify transformations of logarithmic
functions to any base
-
*When we graph logarithmic functions, we will
CHAPTER 5-6
SOLVING EXPONENTIAL AND
LOGARITHMIC EQUATIONS
Objectives:
Use properties and laws of logarithms to
-
simplify and evaluate logarithmic
expressions to any bases.
equations
-
Other bases:
logb v  u
Solving exponential and logarithmic
Solve a variety of application problems by
using exponential and logarithmic
if and only if
bu  v
equations.
Properties:
For b>0 and b  1,
If u=v, then bu  b v for all real numbers b>0.
If u=v, then logb u  logb v , for all real
1. logb v is defined only when v>0
number b>0.
2. logb 1  0  logb b  1
3. log b b  k for every real number k
k
4. b logb v  v for every v>0
CHAPTER 5-7
EXPONENTIAL , LOGARITHMIC , AND
OTHER MODELS
Objectives:
-
For all b, v, w, and k, with b, v, and w positive
and b  1:
Model real data sets with power,
exponential, logarithmic, and logistic
functions.
Product law: logb (vw)  logb v  logb w
Model
Equation
Quotient law: log b ( )  log b v  log b w
v
w
Power
y  ax r
Power law: log b (v k )  k log b v
Exponential
y  ab x  y  aekx
Logarithmic
y  a  b ln x
Logistic
y
For any positive number v,
logb v 
log v
ln v
 log b v 
log b
ln b
a
1  be  kx
MPC4C.01
Date: 2012/1/9
When it has a constant ratio, it will be a
exponential function.
If (x,y) are data points and if the points (x, ln y)
are approximately linear, then an exponential
model may be appropriate for the data.
If (x,y) are data points and if the points (ln x, y)
are approximately linear then a logarithmic
model may be appropriate for the data.
Wyan Chan