Transcript PowerPoint Presentation - 5.1 Integer Exponents and

5.1 Exponential Functions
Rules For Exponents
If a > 0 and b > 0, the following hold true for all real
numbers x and y.
1. a a  a
x
y
x y
x
a
2. y  a x  y
a
 
3. a
x y
a
xy
4. a b  (ab)
x
x
x
x
a
a
5.    x
b
b
6. a0  1
x
1
7. a  x
a
-x
p
q
8. a  a p
q
5.1 Exponential Functions
xxx
1
x


2
5
xxxxx x
x
3
If we apply the quotient rule, we get:
3
x
5
x
x
35
x
2
1
 2
x
5.1 Exponential Functions
•For any nonzero number x:
x
n
1
 n
x
and
1
n

x
n
x
5.1 Exponential Functions
•Examples:
2
2
 2
 3 9
    
 3
 2 4
5
1
3

3

27
3
3
x 252(4)12  15 4 9

x

x

x
2
4
4
x
5.1 Exponential Functions
Examples:
•40
=1
•2-1 = ½
•(½)-2 = 4
•5-2 = 1/25
•(5x-2)3 = 125x-6=125/x6
•(3x/y3)2 = 9x2/y6
•(4x)-1 = 1/(4x)
•(2a3b-3c4)3 = 8a9b-9c12
5.1 Exponential Functions
Simplify:
5
2
4
2
 16  4
 25  5
 3
2
 9 3
 3
5.1 Exponential Functions
 b   b, if
2
b is defined
b b
2
5.1 Exponential Functions
Simplify:
Rewrite:
Notice:

2
x
12
x 
2
6
12  2  6
x
6
5.1 Exponential Functions
Simplify:
Rewrite:
Notice:

2
x
16
x 
2
8
16  2  8
x
8
5.1 Exponential Functions
Simplify:
Rewrite:
Notice:

3
3
x
15
x 
3
5
15  3  5
x
5
5.1 Exponential Functions

If
n
b a
Then
a b
n
Examples
3
 27  3
4
16  2
n
1 1
Since
 3
3
 27
Since
2  16
Since
1 1
4
n
5.1 Exponential Functions
In general, if n is a multiple of m, then
 n m 
m
x x
m
x  x
n
n
n  m 
If n is odd
If n is even
5.1 Exponential Functions
Use the rules for exponents to
solve for x
4x = 128
(2)2x = 27
2x = 7
x = 7/2
•
•
•
•
2x = 1/32
2x = 2-5
x = -5
•
•
•
5.1 Exponential Functions
x
27
•
x3/2y1/3
•
•
•
-x+1
9
=
3
x
2
-x+1
(3 ) = (3 )
33x = 3-2x+2
3x = -2x+ 2
5x = 2
x = 2/5
(x3y2/3)1/2
•
•
•
•
5.1 Exponential Functions
Definition Exponential Function
Let a be a positive real number other than 1,
the function f(x) = ax is the exponential
function with base a.
5.1 Exponential Functions
If b > 1, then the graph of
bx
will:
5
4
3
2
1
y y = 2x
x
•Rise from left to right.
-7 -6 -5 -4 -3 -2 -1
•Not intersect the x-axis.
•Approach the x-axis.
•Have a y-intercept of (0, 1)
-2
-3
-4
-5
1 2 3 4 5 6 7
5.1 Exponential Functions
5
4
3
2
1
y y = (1/ ) x
2
-7 -6 -5 -4 -3 -2 -1
1 2 3 4 5 6 7
If 0 < b < 1, then the graph of
b x will:
•Fall from left to right.
•Not intersect the x-axis.
•Approach the x-axis.
•Have a y-intercept of (0, 1)
-2
-3
-4
-5
x
5.1 Exponential Functions
Natural Exponential Function
where e is the natural base and e
 2.718…
f(x) e
x
x
e 
 1
1  
lim
x
x 
x
5.1 Exponential Functions
Function
f(x) = 2x
h(x) = (0.5)x
g(x) = ex
Domain
(-∞, ∞)
(0, ∞)
(-∞, ∞)
(0, ∞)
(-∞, ∞)
(0, ∞)
Inc.
Dec.
Inc.
Range
Increasing or
Decreasing
Point Shared
On All Graphs
(0, 1)
5.1 Exponential Functions
Use translation of functions
to graph the following.
Determine the domain and
range
f (x) = 2(x + 2) – 3
Domain (-∞, ∞)
Range (-3, ∞)
5.1 Exponential Functions
Definitions Exponential Growth and Decay
The function y = k ax, k > 0 is a model for
exponential growth if a > 1, and a model
for exponential decay if 0 < a < 1.
y  yOb
t
h
y
new amount
yO original amount
b
base
t
time
h
half life
5.1 Exponential Functions
An isotope of sodium, Na, has a half-life of 15
hours. A sample of this isotope has mass 2 g.
(a) Find the amount remaining after t hours.
(b) Find the amount remaining after 60 hours.
• a. y = yobt/h
•
y = 2 (1/2)(t/15)
• b. y = yobt/h
•
y = 2 (1/2)(60/15)
•
y = 2(1/2)4
•
y = .125 g
5.1 Exponential Functions
A bacteria double every three days. There are
50 bacteria initially present
(a) Find the amount after 2 weeks.
(b) When will there be 3000 bacteria?
• a. y = yobt/h
•
y = 50 (2)(14/3)
•
y = 1269 bacteria
5.1 Exponential Functions
A bacteria double every three days. There are
50 bacteria initially present
When will there be 3000 bacteria?
• b. y = yobt/h
•
3000 = 50 (2)(t/3)
•
60 = 2t/3
•
5.2 Simple and Compound
Interest
Formulas for Simple Interest
Suppose P dollars are invested at a simple interest
rate r, where r is a decimal, then P is called the
principal and P ·r is the interest received at the
end of one interest period.
T  P(1  nr)
5.2 Simple and Compound
Interest
Formulas for Compound Interest
After t years, the balance A in an account with principal
P and annual interest rate r is given by the two
formulas below.
 r
1. For n compoundings per year: A  P1  
 n
2. For continuous compounding: A  Pert
nt
5.2 Simple and Compound
Interest
Find the balance after 10 years if $1000.00 is invested
at 4% and the account pays simple interest.
T  P(1  nr)
T  1000[1  10(.04)]
T  $1400 .00
5.2 Simple and Compound
Interest
Find the balance after 10 years if $1000.00 is invested
at 4% and the interest is compounded:
a. Semiannually
b. Monthly:
c. Continuously:
 r
A  P1  
 n
 r
A  P1  
 n
A  Pert
nt
nt
 .04 
A  10001 

2 

( 2*10 )
 .04 
A  10001 

 12 
A  1000e(.04*10)
$1485.95
(12*10 )
$1490.83
$1491.82
5.3 Effective Rate and
Annuities
Effective Annual Rate
The effective annual rate of ieff of APR
compounded k times per year is given by the
k
equation
r


ieff  1    1
 k
Another name for effective annual rate is
effective yield
5.3 Effective Rate and
Annuities
What is the better rate of return,
7% compounded quarterly or 7.2 %
compounded semianually?
k
 r
ieff  1    1
 k
5.3 Effective Rate and
Annuities
4
 .07 
ieff  1 
 1
4 

2
 .072
ieff  1 
 1
2 

1.071 – 1 =
.071 = 7.1%
1.073 – 1 =
.073 = 7.3%
7.2% compounded semiannually is better.
5.3 Effective Rate and
Annuities
What is the better rate of return,
8 % compounded monthly or 8.2 %
compounded quarterly?
k
 r
ieff  1    1
 k
5.3 Effective Rate and
Annuities
12
 .08 
ieff  1 
 1
 12 
4
 .082
ieff  1 
 1
4 

8.2% quarterly is better.
8.3%
8.5%
5.3 Effective Rate and
Annuities
Future Value of an Ordinary Annuity
The Future Value S of an ordinary annuity
consisting of n equal payments of R dollars,
each with an interest rate i per period is
(1  i )  1
SR
i
n
5.3 Effective Rate and
Annuities
Suppose $25.00 per month is invested
at 8% compounded quarterly. How much will
be in the account after one year?
•1st quarter
•2nd quarter
•3rd quarter
•4th quarter
$25.00
$25.00(1+.08/4)+ $25.00 = $50.50
$50.50(1+.08/4)+ $25.00 = $76.51
$76.51(1+.08/4) + $25.00 = $103.04
5.3 Effective Rate and
Annuities
Present Value of an Ordinary Annuity
The Present Value A of an ordinary
annuity consisting of n equal payments of R
dollars, each with an interest rate i per period is
1  (1  i )
A R
i
n
5.4 Logarithmic Functions
The inverse of an exponential
function is called a logarithmic
function.
y
Definition: x = a if and only if
y = log a x
5.4 Logarithmic Functions
•log 4 16 = 2 ↔ 42 = 16
4
•log 3 81 = 4 ↔ 3 = 81
•log10 100 = 2 ↔ 102 = 100
5.4 Logarithmic Functions
Sketch a graph of f (x) = 2x and sketch a graph of
its inverse. What is the domain and range of the
inverse of f.
Domain: (0, ∞)
Range: (-∞, ∞)
5.4 Logarithmic Functions
The function f (x) = log a x is called a logarithmic
function.
Domain: (0, ∞)
Range: (-∞, ∞)
Asymptote: x = 0
Increasing for a > 1
Decreasing for 0 < a < 1
Common Point: (1, 0)
•
•
•
•
5.4 Logarithmic Functions
Find the inverse of g(x) = 3x.
1
g ( x)  log3 x
(1,3)
(0,1)
(-1,1/3)
Note: The function and
it’s inverse are symmetrical
about the line y = x.
(3,1)
(1,0)
(1/3,-1)
5.4 Logarithmic Functions
Find the inverse of g(x) = ex.
1
g ( x)  loge x  ln x
ln x is called the natural
logarithmic function
5.4 Logarithmic Functions
b 1
So
logb 1  0
log3 1  0
b b
So
logb b  1
log10  1
So
logb b  x
ln e  3
0
1
b b
x
x
logb x  logb x
x
So
b
logb x
3
x
log 5
10
5
5.4 Logarithmic Functions
1.
2.
3.
4.
5.
loga(ax) = x for all x  
alog ax = x for all x > 0
loga(xy) = logax + logay
loga(x/y) = logax – logay
logaxn = n logax
Common Logarithm: log 10 x = log x
Natural Logarithm: log e x = ln x
All the above properties hold.
5.4 Logarithmic Functions
Product
Rule
logb m  n  logb m  logb n
log3 36  log3 9  4
 log3 9  log3 4  2  log3 4
5.4 Logarithmic Functions
Quotient
Rule
m
logb    logb m  logb n
n
 50 
log5 50  log5 2  log5  
 2
 log5 25  2
5.4 Logarithmic Functions
Power
Rule
logb m  c  logb m
c
ln e  4  ln e
4
 4 1  4
5.4 Logarithmic Functions
Expand
x y
log 5 
 z
3
2



 (log5 x  log5 y )  log5 z
3
2
 (3 log5 x  2 log5 y)  log5 z
5.4 Logarithmic Functions
Find an equation of best fit for the data
(1,3), (2,12), (3,27), (4,48)
y  axm
ln 3  ln a
ln 4  ln 2m
ln y  ln axm
a3
m2
ln y  ln a  ln x m
ln y  ln 3  m ln x
y  3x2
ln y  ln a  m ln x
ln 12  ln 3  m ln 2
ln 3  ln a  m ln 1
ln 12  ln 3  m ln 2
5.5 Graphs of Logarithmic
Functions
The function f (x) = log a x is called a logarithmic
function.
Domain: (0, ∞)
Range: (-∞, ∞)
Asymptote: x = 0
Increasing for a > 1
Decreasing for 0 < a < 1
Common Point: (1, 0)
•
•
•
•
5.5 Graphs of Logarithmic
Functions
The natural and common logarithms can be
found on your calculator. Logarithms of other
bases are not. You need the change of base
formula.
log x
loga x 
b
logb a
where b is any other appropriate base.
(usually base 10 or base e)
5.5 Graphs of Logarithmic
Functions
Sketch the graph of
3
0
y  log3 ( x  2)
5
1
Domain (2,)
Range (-, )
29 3
11 2
5.5 Graphs of Logarithmic
Functions
Sketch the graph of
y  log2 (2 x  4)
y  log2 2( x  2)
y  log2 2  log2 ( x  2)
y  1  log2 ( x  2)
Domain (-2,)
Range (-, )
5.5 Graphs of Logarithmic
Functions
Sketch the graph of
y  1  ln(x  3)
Domain (-3,)
Range (-, )
5.5 Graphs of Logarithmic
Functions
On the Richter scale, the magnitude R of an
earthquake can be measured by the intensity
model.
R = Magnitude
a
R  log   B a = Amplitude
T = Period
T 
B = Damping Factor
5.5 Graphs of Logarithmic
Functions
What is the magnitude on the Richter scale of an
earthquake if a = 300, T = 30 and B = 1.2?
a
R  log   B
T 
R  log10  1.2
 300
R  log
  1.2
 30 
R  1 1.2
R  2.2
5.6 Solving Exponential
Equations
Solve: 4 3x = 16 x – 2
 The bases can be rewritten as:

(22) 3x = (24) (x – 2)
6x = 2 4x – 8
2
 6x = 4x – 8
 2x = -8

x = -4

5.6 Solving Exponential
Equations


To solve exponential equations, pick a
convenient base (often base 10 or base e)
and take the log of both sides.
Solve:
4  21
3x
5.6 Solving Exponential
Equations
4  21
3x
•Take the log of
both sides:
•Power rule:
   log 21
3x
log 4
3xlog 4  log 21
5.6 Solving Exponential
Equations
3xlog 4  log 21
•Solve for x:
x  3 log 4  log 21
•Divide:
log 21
x
3 log 4
x  0.732
5.6 Solving Exponential
Equations


To solve logarithmic equations, write both
sides of the equation as a single log with the
same base, then equate the arguments of the
log expressions.
Solve:
log2x  8  logx  1  logx  2
5.6 Solving Exponential
Equations
log2x  8  logx  1  logx  2
•Write the left side as a single logarithm:
 2x  8 
log
  logx  2
 x 1 
5.6 Solving Exponential
Equations
 2x  8 
log
  logx  2
 x 1 
•Equate the arguments:
 2x  8 

 x2
 x 1 
5.6 Solving Exponential
Equations
•Solve for x:
 2x  8 

 x2
 x 1 
 2x  8 
x  1
  x  2x  1
 x 1 
5.6 Solving Exponential
Equations
2 x  8  x  2x  1
2 x  8  x  3x  2
2
0  x  x6
2
0  x  3x  2
5.6 Solving Exponential
Equations
0  x  3x  2
x  3, 2
•Check for extraneous solutions.
log2x  8  logx  1  logx  2
•x = -3, since the argument of a log
cannot be negative
5.6 Solving Exponential
Equations


To solve logarithmic equations with one side
of the equation equal to a constant, change
the equation to an exponential equation
Solve:
log x  log4x  6
2
5.6 Solving Exponential
Equations
log x  log4x  6
2
•Write the left side as a single logarithm:


log4 x   6
log x  4x  6
2
3
5.6 Solving Exponential
Equations
 
log 4 x  6
3
•Write as an exponential equations:
4 x  10
3
6
5.6 Solving Exponential
Equations
4 x  10
3
6
•Solve for x:
x  250000
3
x  62.99