PowerPoint Presentation - 5.1 Integer Exponents and
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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
xxx
1
x
2
5
xxxxx x
x
3
If we apply the quotient rule, we get:
3
x
5
x
x
35
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 252(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 P1
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 P1
n
r
A P1
n
A Pert
nt
nt
.04
A 10001
2
( 2*10 )
.04
A 10001
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
SR
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
a3
m2
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
3xlog 4 log 21
5.6 Solving Exponential
Equations
3xlog 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:
log2x 8 logx 1 logx 2
5.6 Solving Exponential
Equations
log2x 8 logx 1 logx 2
•Write the left side as a single logarithm:
2x 8
log
logx 2
x 1
5.6 Solving Exponential
Equations
2x 8
log
logx 2
x 1
•Equate the arguments:
2x 8
x2
x 1
5.6 Solving Exponential
Equations
•Solve for x:
2x 8
x2
x 1
2x 8
x 1
x 2x 1
x 1
5.6 Solving Exponential
Equations
2 x 8 x 2x 1
2 x 8 x 3x 2
2
0 x x6
2
0 x 3x 2
5.6 Solving Exponential
Equations
0 x 3x 2
x 3, 2
•Check for extraneous solutions.
log2x 8 logx 1 logx 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:
log4 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